Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
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Theorem 3.32. Let \( \alpha \) be an irrational number with convergents \( {p}_{\nu }/{q}_{\nu } \) defined by (3.24) and (3.25). Let \( q = \mathop{\sum }\limits_{{\nu = k}}^{\infty }{d}_{\nu }{q}_{\nu } \) be the \( \alpha \) -adic expansion of the positive integer \( q \), as in Lemma 3.28. Assume \( k \geq 2 \), or... | Proof. By Lemma 3.28, the quantity \( {q\alpha } - p \) lies between \( {\left( -1\right) }^{k}/{q}_{k + 2}^{\prime } \) and \( {\left( -1\right) }^{k}/{q}_{k}^{\prime } \) . Under the given conditions on \( k,{q}_{k}^{\prime } > {q}_{k} \geq 2 \) . Hence \( x = \) \( {2\pi i}\left( {{q\alpha } - p}\right) \) is less t... | Yes |
Theorem 3.34. For every integer \( k \geq 0 \) (or \( k \geq 1 \) if \( {a}_{1} = 1 \) ), there exists a complex root \( \omega \) of \( f \) of the form\n\n\[ \omega = D + {2\pi i}\frac{{q}_{k}}{{w}_{1}} - {2\pi i}{\left( -1\right) }^{k}\frac{{m}_{2}{e}^{-{w}_{2}D}}{{f}^{\prime }\left( D\right) {q}_{k + 1}^{\prime }} ... | Proof. In this case, \( q = {q}_{k} \) is the \( \alpha \) -adic expansion of \( q \) . Put \( p = {p}_{k} \) . Then \( x = {2\pi i}{\left( -1\right) }^{k}/{q}_{k + 1}^{\prime } \), which is less than \( \pi \) in absolute value. The rest of the proof is the same as the proof of Theorem 3.32. | No |
Lemma 3.39. Let \( 0 < {w}_{1} < {w}_{2} < \cdots < {w}_{M} \), let \( D \) be the real number such that \( \mathop{\sum }\limits_{{j = 1}}^{M}{m}_{j}{e}^{-{w}_{j}D} = 1 \), and let \( \Delta = \Delta \left( {{x}_{2},\ldots ,{x}_{M}}\right) \) be implicitly defined by\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{M}{m}_{j}{e}... | Proof. The proof is analogous to that of Lemma 3.29. Taking the derivative with respect to \( {x}_{k}\left( {k \geq 2}\right) \) gives\n\n\[ {m}_{k}{e}^{-{w}_{k}D} + {f}^{\prime }\left( D\right) \frac{\partial \Delta }{\partial {x}_{k}}\left( {0,\ldots ,0}\right) = 0. \]\n\nWe thus find the coefficients of the linear p... | Yes |
Theorem 3.41. Let \( M \geq 2 \) and let \( {w}_{1},\ldots ,{w}_{M} \) be weights of a nonlattice equation. Let \( Q \) and \( q \) be as in Lemma 3.16. Then \( f \) has a complex root close to \( D + {2\pi iq}/{w}_{1} \) at a distance of at most \( O\left( {Q}^{-2}\right) \) from the line \( \operatorname{Re}s = D \) ... | Proof. Again, for \( j = 2,\ldots, M \), the numbers \( {x}_{j} \) are purely imaginary, so the terms of degree 1 (and of every odd degree) give a correction in the imaginary direction, and only the terms of even degree will give a correction in the real direction. Since \( \left| {x}_{j}\right| < {2\pi }/Q \), the the... | No |
One of the simplest nonlattice strings is the golden string, introduced in Section 2.3.5. It is the nonlattice string with \( M = 2 \) and \( \alpha = \phi ,{w}_{1} = \log 2 \) . The continued fraction of the golden ratio is\n\n\[ \phi = \frac{1 + \sqrt{5}}{2} = \left\lbrack \left\lbrack {1,1,1,\ldots }\right\rbrack \r... | Numerically, we find \( D \approx {.7792119034} \) and the following approximation to the power series \( \Delta \left( x\right) \) :\n\n\[ - {.47862x} + {.08812}{x}^{2} + {.00450}{x}^{3} - {.00205}{x}^{4} - {.00039}{x}^{5} + \ldots \]\n\nFor every \( k \geq 0 \), we find a complex dimension close to \( D + {2\pi i}{q}... | Yes |
Figure 3.10 gives the complex dimensions and the density of their real parts of the generic nonlattice string with weights \( {w}_{1} = \log 2 \) and \( {w}_{2} = \alpha \log 2 \), where \( \alpha \) is the positive real number with continued fraction \( \left\lbrack \left\lbrack {1,2,3,4,\ldots }\right\rbrack \right\r... | One can compute that\n\n\[ \alpha = \frac{\mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{1}{{\left( n!\right) }^{2}}}{\mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{1}{n!\left( {n + 1}\right) !}} = \frac{{I}_{0}\left( 2\right) }{{I}_{1}\left( 2\right) } \]\n\nwhere \( {I}_{k}\left( z\right) = \mathop{\sum }\limits_{{n ... | Yes |
Corollary 3.49. Let \( M \geq 2 \) . The best dimension-free region that \( \mathcal{L} \) can have is of size\n\n\[ \left\{ {\sigma + {it} \in \mathbb{C} : \sigma \geq D - O\left( {t}^{-2/\left( {M - 1}\right) }\right) }\right\} \]\n\nwhere the implied constant is positive and depends only on \( {w}_{1},\ldots ,{w}_{M... | If \( {w}_{1},\ldots ,{w}_{M} \) is b-approximable, then the dimension-free region has the form\n\n\[ \left\{ {\sigma + {it} \in \mathbb{C} : \sigma \geq D - O\left( {{b}^{-2}\left( {{w}_{1}t/{2\pi }}\right) {t}^{-2/\left( {M - 1}\right) }}\right) }\right\} ,\]\n\nwhere the \( O \) -term is a positive function, bounded... | Yes |
Theorem 3.53. The set of real parts of the complex dimensions of a non-lattice string has no isolated points. | Proof. By Remark 3.40, every complex dimension gives rise to a sequence of complex dimensions close to the points \( \omega + {2\pi iq}/{w}_{1} \), for integers \( q \) . Since the corrective terms are not all purely imaginary, we find complex dimensions with real parts close to \( \operatorname{Re}\omega \) . When \( ... | Yes |
We compute in two different ways the frequencies of the prime string \( \mathfrak{P} \), defined by (4.13). | According to Equations (4.15) and (4.17), we have\n\n\[ \n{\zeta }_{\nu }\left( s\right) = {\zeta }_{\mathfrak{P}}\left( s\right) \cdot \zeta \left( s\right) = - \frac{{\zeta }^{\prime }\left( s\right) }{\zeta \left( s\right) } \cdot \zeta \left( s\right) = - {\zeta }^{\prime }\left( s\right) , \n\] \n\nwhere \( {\zeta... | Yes |
Theorem 4.9. The measure \( {\mu }_{\mathcal{L}} \) defined by (4.38) satisfies the following scaling property, which we call its self-similarity property:\n\n\[ \n{\mu }_{\mathcal{L}}\left( A\right) = {\delta }_{1 \in A} + \mathop{\sum }\limits_{{j = 1}}^{N}{\mu }_{\mathcal{L}}\left( {{r}_{j}A}\right) \n\]\n\n(4.40)\n... | Proof of Theorem 4.9. The measure of \( A \) is\n\n\[ \n{\mu }_{\mathcal{L}}\left( A\right) = \mathop{\sum }\limits_{{\mathbf{e} \geq 0}}\left( \begin{matrix} \sum \mathbf{e} \\ \mathbf{e} \end{matrix}\right) {\delta }_{{\mathbf{r}}^{-\mathbf{e}} \in A} \n\]\n\n(4.41)\n\nand that of \( {r}_{j}A \) is\n\n\[ \n{\mu }_{\m... | Yes |
The solution space of the recursion relation\n\n\\[ \n{a}_{n} = \\mathop{\\sum }\\limits_{{j = 1}}^{N}{a}_{n - {k}_{j}}\\;\\left( {n \\in \\mathbb{Z}}\\right) \n\\]\n\nhas dimension \\( {k}_{N} \\) . For each complex solution \\( z \\) of the polynomial equation\n\n\\[ \n{z}^{{k}_{N}} = \\mathop{\\sum }\\limits_{{j = 1... | Alternatively, for every \\( t \\in \\mathbb{R} \\), we obtain \\( m\\left( z\\right) \\) solutions\n\n\\[ \nn \\mapsto {\\left( n + t\\right) }^{q}{z}^{n + t}\\;\\left( {n \\in \\mathbb{Z}}\\right) .\n\\]\n\nThese solutions (for fixed \\( t \\), and all \\( z \\) and \\( q,0 \\leq q \\leq m\\left( z\\right) - 1 \\) ) ... | Yes |
Lemma 5.9 (Truncated pointwise formula). Let \( k \geq 1 \) be an integer and let \( \eta \) be a generalized fractal string. Then, for all \( x > 0 \) and \( n \geq 1 \), the function \( {N}_{\eta }^{\left\lbrack k\right\rbrack }\left( x\right) \) is approximated by \[ \mathop{\sum }\limits_{{\omega \in {\mathcal{D}}_... | Proof. The proof is given in two steps. The first step consists of deriving an approximate expression for \( {N}_{\eta }^{\left\lbrack k\right\rbrack }\left( x\right) \) . For this, we consider the line integral \[ J\lef | No |
Theorem 5.10 (The pointwise explicit formula, with error term). Let \( \eta \) be a languid \( {}^{4} \) generalized fractal string, and let \( k \) be an integer such that \( k > \max \{ 1,\kappa + 1\} \), where \( \kappa \) is the exponent occurring in the statement of \( \mathbf{{L1}} \) and \( \mathbf{{L2}} \) . Th... | Here, for \( x > 0, R\left( x\right) = {R}_{\eta }^{\left\lbrack k\right\rbrack }\left( x\right) \) is the error term, given by the absolutely convergent integral \n\n\[ \nR\left( x\right) = {R}_{\eta }^{\left\lbrack k\right\rbrack }\left( x\right) = \frac{1}{2\pi i}{\int }_{S}{x}^{s + k - 1}{\zeta }_{\eta }\left( s\ri... | Yes |
Theorem 5.14 (The pointwise formula, without error term). Let \( \eta \) be a generalized fractal string satisfying hypotheses \( \mathbf{L1} \) and \( \mathbf{L2}^{\prime} \) ; i.e., \( \eta \) is strongly languid (see Definition 5.3 and Equations (5.19) and (5.21) above). Let \( k \) be a positive integer such that \... | Proof. For a fixed integer \( n \geq 1 \), we apply Lemma 5.9 with the screen \( S_{m} \) given by hypothesis \( \mathbf{L2}^{\prime} \). We now assume that \( k > \kappa \) (instead of \( k > \kappa + 1 \) as in the proof of Theorem 5.10). We first let \( m \) tend to \( \infty \), while keeping \( n \) fixed. Since t... | Yes |
Theorem 5.17. Let\n\n\[ \mathop{\sum }\limits_{{\operatorname{Re}\omega < D}}{a}_{\omega }{x}^{\omega }{\left( \log x\right) }^{{m}_{\omega }} \]\n\n(5.41)\n\nbe an absolutely convergent sum over the visible complex dimensions with real part less than \( D \), arising from Theorem 5.10 or 5.14 (hence \( {a}_{\omega } \... | Proof. The (conditional) convergence of the sum over the complex dimensions with real part \( D \) follows from Theorem 5.10 or 5.14, and the assumption that (5.41) converges absolutely.\n\nIt remains to estimate the sum in (5.41). This is done by adapting the method of [In, pp. 82 and 33] as follows. Let \( \varepsilo... | Yes |
Theorem 5.18 (The distributional formula, with error term). Let \( \eta \) be a languid generalized fractal string; i.e., it satisfies hypotheses L1 and L2 (see Equations (5.19) and (5.20) above). Then, for every \( k \in \mathbb{Z} \), the distribution \( {\mathcal{P}}_{\eta }^{\left\lbrack k\right\rbrack } \) is give... | Proof of Theorem 5.18. First, given an integer \( n \geq 1 \), we apply Lemma 5.9 for \( k > \kappa + 1 \) . Let \( \varphi \) be a test function. Then\n\n\[ \n\left\langle {{\mathcal{P}}_{\eta }^{\left\lbrack k\right\rbrack },\varphi }\right\rangle = {\int }_{0}^{\infty }{N}_{\eta }^{\left\lbrack k\right\rbrack }\left... | Yes |
Theorem 5.22 (The distributional formula, without error term). Let \( \eta \) be a generalized fractal string that is strongly languid; i.e., it satisfies hypotheses \( \mathbf{L1} \) and \( \mathbf{L2} \) ’ (see Equations (5.19) and (5.21) above). Then, for every \( k \in \mathbb{Z} \) and for test functions with comp... | Proof. Again, given \( n \geq 1 \), we apply Lemma 5.9, but now for \( k > \kappa \) (instead of \( k > \kappa + 1 \) as in the proof of Theorem 5.18). Let \( \varphi \) be a test function whose support is contained in \( \lbrack A + \delta ,\infty ) \), for some \( \delta > 0 \) . Then \( \widetilde{\varphi }\left( s\... | Yes |
Corollary 5.23 (The density of states formula). Under the same hypotheses as in Theorem 5.18 (respectively, Theorem 5.22), we have the following distributional explicit formula for \( {\mathcal{P}}_{\eta }^{\left\lbrack 0\right\rbrack } = \eta \) : | \[ \eta = \mathop{\sum }\limits_{{\omega \in {\mathcal{D}}_{\eta }\left( W\right) }}\operatorname{res}\left( {\frac{{x}^{s + k - 1}{\zeta }_{\eta }\left( s\right) }{{\left( s\right) }_{k}};\omega }\right) + {\mathcal{R}}_{\eta }^{\left\lbrack 0\right\rbrack }, \] (5.49) where \( {\mathcal{R}}_{\eta }^{\left\lbrack 0\ri... | Yes |
Lemma 5.25. Let \( f\left( s\right) \) and \( g\left( s\right) \) be meromorphic in a disc around 0 . Then the function \( z \mapsto \operatorname{res}\left( {f\left( {s - z}\right) g\left( s\right) ;0}\right) + \operatorname{res}\left( {f\left( {s - z}\right) g\left( s\right) ;z}\right) \) is holomorphic in the same d... | Proof. This follows since the function can be written as the integral over a small circle around \( s = 0 \) and \( s = z \) of \( f\left( {s - z}\right) g\left( s\right) \), and this function is analytic in \( z \) . | No |
Theorem 5.27 (Extended distributional formula, without error term). Let \( \eta \) be a strongly languid generalized fractal string (see Definition 5.3). Let \( k \in \mathbb{Z} \) and let \( q \in \mathbb{N} \) be such that \( k + q > \max \{ 1,\kappa \} \), where \( \kappa \) is given as in (5.19) and (5.21). Further... | Then formula (5.54), with \( {\mathcal{R}}_{\eta }^{\left\lbrack k\right\rbrack } \equiv 0 \), gives the distributional explicit formula without error term at level \( k \) for \( \varphi \) .\n\nProof of Theorems 5.26 and 5.27. First of all, the condition at infinity on \( \varphi \) implies that \( \left\langle {{\ma... | Yes |
Theorem 5.30 (Order of the distributional error term). Fix \( k \in \mathbb{Z} \). Assume that the hypotheses of Theorem 5.18 (or more generally, of Theorem 5.26, with \( k + q > \kappa + 1 \)) are satisfied. Then the distribution \( {\mathcal{R}}_{\eta }^{\left\lbrack k\right\rbrack } \), given by (5.46), is of asympt... | Proof. The integral (5.46) for \( \left\langle {{\mathcal{R}}_{\eta }^{\left\lbrack k\right\rbrack },\varphi }\right\rangle \) converges absolutely. Let \( \varphi \) be a test function with compact support. When we replace \( \varphi \) by \( {\varphi }_{a} \) in (5.46), we see, by formula (5.61), that the absolute va... | Yes |
Theorem 5.31. Let \( v \leq D \) . Assume that the hypotheses of Theorem 5.18 (or of Theorem 5.26, with \( k + q > \kappa + 1 \) ) are satisfied, with a screen contained in the open half-plane \( \operatorname{Re}s < v \) . Assume, in addition, that there exists a screen \( {S}_{0} \) contained in \( \operatorname{Re}s... | Proof. We write this distribution as \( {\mathcal{R}}_{0,\eta }^{\left\lbrack k\right\rbrack }\left( x\right) - {\mathcal{R}}_{\eta }^{\left\lbrack k\right\rbrack }\left( x\right) \), where \( {\mathcal{R}}_{0,\eta }^{\left\lbrack k\right\rbrack } \) is the error term associated with the screen \( {\dot{S}}_{0} \) . Th... | Yes |
Recall the definition of the continued fraction of a real number \( \alpha > 1 \) given in Section 3.5.1. We construct a nonlattice self-similar string \( \mathcal{L} \) with two scaling ratios \( {r}_{1} = {e}^{-1},{r}_{2} = {e}^{-\alpha } \), where \( \alpha \) will be specified below. \( {}^{11} \) Consider the func... | Indeed, we have \( {e}^{-\left( {D + {2\pi i}{q}_{n}}\right) } = {e}^{-D} \) and\n\n\[ \n{e}^{-\left( {D + {2\pi i}{q}_{n}}\right) \alpha } = {e}^{-{D\alpha }}{e}^{-{2\pi i}{q}_{n}\alpha + {2\pi i}{p}_{n}} = {e}^{-{D\alpha }}{e}^{{2\pi i}{\left( -1\right) }^{n}/{q}_{n + 1}^{\prime }}.\n\]\n\nHence, by the same techniqu... | Yes |
Consider the measure \( \mu \) on \( \lbrack 1,\infty ) \) defined by\n\n\[ \mu = \mathop{\sum }\limits_{{n \in \mathbb{Z}\smallsetminus \{ 0\} }}\frac{{x}^{D - 1 - {d}_{n} + {in}}}{{n}^{2}}{dx} \]\n\nwhere the real numbers \( {d}_{n} = {d}_{-n} \) are small, and will be specified below. The geometric zeta function of ... | Therefore, we have\n\n\[ {\int }_{D + i\left( {n - 1/2}\right) }^{D + i\left( {n + 1/2}\right) }\left| {{\zeta }_{\mu }\left( {D + {it}}\right) }\right| {t}^{-\kappa }{dt} \approx - \frac{\log {d}_{n}}{{n}^{2 + \kappa }}, \]\n\nas \( n \rightarrow \infty \) .\n\nWe conclude that if \( {d}_{n} = \exp \left( {-{n}^{n}}\r... | Yes |
The generalized fractal string\n\n\\[ \eta = \\mathop{\\sum }\\limits_{{n = 1}}^{\\infty }{\\left( -1\\right) }^{n - 1}{\\delta }_{\\{ n\\} } \\]\n\nhas no complex dimensions. | Indeed, by [Ti, Eq. (2.2.1), p. 16], the associated geometric zeta function is\n\n\\[ {\\zeta }_{\\eta }\\left( s\\right) = \\mathop{\\sum }\\limits_{{n = 1}}^{\\infty }{\\left( -1\\right) }^{n - 1}{n}^{-s} = \\left( {1 - {2}^{1 - s}}\\right) \\zeta \\left( s\\right) . \\]\n\nSince the pole of \\( \\zeta \\left( s\\rig... | Yes |
Theorem 6.1. Let \( \omega \) be a complex dimension of multiplicity \( m \) and let the principal part of \( {\zeta }_{\eta }\left( s\right) \) at \( s = \omega \) be given by formula (6.2). Then the local term (6.1) in the explicit formula at \( \omega \) is given by | \[ {x}^{\omega - 1}\mathop{\sum }\limits_{{j = 1}}^{m}{a}_{j}\frac{{\left( \log x\right) }^{j - 1}}{\left( {j - 1}\right) !} \] (6.7) for \( k = 0 \), and by \[ {x}^{\omega + k - 1}\mathop{\sum }\limits_{{j = 1}}^{m}{a}_{j}\mathop{\sum }\limits_{{\iota = 0}}^{{k - 1}}\mathop{\sum }\limits_{{\mu = 0}}^{{j - 1}}\frac{{\l... | Yes |
Theorem 6.20. Let \( a > 0 \) and let \( \mathcal{L} \) be the ordinary fractal string with lengths \( {l}_{j} \) given by (6.63). Then \( {\zeta }_{\mathcal{L}}\left( s\right) = \mathop{\sum }\limits_{{j = 1}}^{\infty }{l}_{j}^{s} \) has a meromorphic continuation to all of \( \mathbb{C} \) . The poles of \( {\zeta }_... | Proof. We compute the first term of an asymptotic expansion of \( {l}_{j} \) :\n\n\[ \n{l}_{j} = {j}^{-a} - {\left( j + 1\right) }^{-a} = a{\int }_{j}^{j + 1}{x}^{-a - 1}{dx} = a{j}^{-a - 1} + H\left( j\right) , \n\] \n\nwhere \( H\left( j\right) = a{\int }_{j}^{j + 1}\left( {{x}^{-a - 1} - {j}^{-a - 1}}\right) {dx} \)... | Yes |
Theorem 6.25. The Sierpinski drum has oscillations of order \( D \) in its spectrum, where \( D = {\log }_{2}3 \) is the Minkowski dimension of the Sierpinski gasket. | Proof. The fact that \( G \) is nonconstant follows from Theorem 11.12 below, according to which the Dirichlet series \( {\zeta }_{\mathcal{T}}\left( s\right) \) does not have an infinite vertical sequence of zeros in arithmetic progression. It follows that the expansion (6.86), interpreted distributionally (as, for ex... | No |
Theorem 7.8 (Euler sum). For \( \operatorname{Re}s > D \), we have the following expression for the logarithmic derivative of \( {\zeta }_{\mathfrak{w}} \) :\n\n\[ - \frac{{\zeta }_{\mathfrak{w}}^{\prime }}{{\zeta }_{\mathfrak{w}}}\left( s\right) = \mathop{\sum }\limits_{{\mathfrak{p} \in \sigma \smallsetminus \sum }}\... | Proof. We write the sum in (7.4) over the finite words \( \mathfrak{x} \) as a sum over the primitive words and repetitions of these. An orbit \( \mathfrak{p} \) contains \( \# \mathfrak{p} \) different primitive words of length \( \# \mathfrak{p} \), hence we obtain\n\n\[ \mathop{\sum }\limits_{\mathfrak{x}}\frac{{\ma... | Yes |
Corollary 7.9 (Euler product). The function \( {\zeta }_{\mathfrak{w}}\left( s\right) \) has the following expansion as a product over all primitive periodic orbits \( \mathfrak{p} \) of \( {\mathcal{F}}_{\mathfrak{w}} \) :\n\n\[ \n{\zeta }_{\mathfrak{w}}\left( s\right) = \mathop{\prod }\limits_{{\mathfrak{p} \in \sigm... | Proof. In (7.7), we sum over \( k \) to obtain\n\n\[ \n\frac{{\zeta }_{\mathfrak{w}}^{\prime }}{{\zeta }_{\mathfrak{w}}}\left( s\right) = - \mathop{\sum }\limits_{{\mathfrak{p} \in \sigma \smallsetminus \sum }}\frac{{\mathfrak{w}}_{\mathrm{{tot}}}\left( \mathfrak{p}\right) {e}^{-s{\mathfrak{w}}_{\mathrm{{tot}}}\left( \... | Yes |
Corollary 7.12. We have the following relation between \( {\zeta }_{\mathfrak{w}}^{\prime }/{\zeta }_{\mathfrak{w}} \) and \( {\psi }_{\mathfrak{w}} \) :\n\n\[ \n- \frac{{\zeta }_{\mathfrak{w}}^{\prime }}{{\zeta }_{\mathfrak{w}}}\left( s\right) = {\int }_{0}^{\infty }{x}^{-s}d{\psi }_{\mathfrak{w}}\left( x\right) \n\] ... | The integral on the right-hand side of (7.11) is a Riemann-Stieltjes integral associated with the monotonic function \( {\psi }_{\mathfrak{w}} \) . | No |
The dynamical zeta function associated with a self-similar flow has a meromorphic continuation to the whole complex plane, given by\n\n\[ \n{\zeta }_{\mathfrak{w}}\left( s\right) = \frac{1}{1 - \mathop{\sum }\limits_{{j = 1}}^{N}{r}_{j}^{s}}. \n\] | Proof. The sum over periodic words of fixed length \( l \) can be computed as follows:\n\n\[ \n\mathop{\sum }\limits_{{\mathfrak{x} : l\left( \mathfrak{x}\right) = l}}r{\left( \mathfrak{x}\right) }^{s} = \mathop{\sum }\limits_{{{a}_{1} = 1}}^{N}\mathop{\sum }\limits_{{{a}_{2} = 1}}^{N}\cdots \mathop{\sum }\limits_{{{a}... | Yes |
Example 7.24 (The Cantor flow). This is the self-similar flow on the alphabet \( \{ 0,1\} \), with two equal weights \( {w}_{1} = {w}_{2} = \log 3 \) . It has \( {2}^{n} \) periodic words \( {}^{1} \) of weight \( n\log 3 \), for \( n = 1,2,\ldots \) . The dynamical zeta function of this flow is given by | \[ {\zeta }_{\mathrm{{CF}}}\left( s\right) = \frac{1}{1 - 2 \cdot {3}^{-s}}. \] (7.21) The logarithmic derivative equals \[ - \frac{{\zeta }_{\mathrm{{CF}}}^{\prime }}{{\zeta }_{\mathrm{{CF}}}}\left( s\right) = 2\log 3 \cdot \frac{{3}^{-s}}{1 - 2 \cdot {3}^{-s}}. \] (7.22) | Yes |
The Fibonacci flow is the flow Fib on the alphabet \( \{ 0,1\} \) with weights \( {w}_{1} = \log 2,{w}_{2} = 2\log 2 \) . Its periodic words have weight \( \log 2,2\log 2,\ldots, n\log 2,\ldots \), with multiplicity respectively \( 1,2,\ldots ,{F}_{n + 1},\ldots \), the Fibonacci numbers defined by (2.19). The dynamica... | \[ {\zeta }_{\mathrm{{Fib}}}\left( s\right) = \frac{1}{1 - {2}^{-s} - {4}^{-s}} \] (7.23) with logarithmic derivative \[ - \frac{{\zeta }_{\mathrm{{Fib}}}^{\prime }}{{\zeta }_{\mathrm{{Fib}}}}\left( s\right) = \log 2 \cdot \frac{{2}^{-s} + 2 \cdot {4}^{-s}}{1 - {2}^{-s} - {4}^{-s}}. \] (7.24) | Yes |
Example 7.26 (The golden flow). We consider the nonlattice flow GF with weights \( {w}_{1} = \log 2 \) and \( {w}_{2} = \phi \log 2 \), where \( \phi = \left( {1 + \sqrt{5}}\right) /2 \) is the golden ratio. We call it the golden flow. Its dynamical zeta function is \[ {\zeta }_{\mathrm{{GF}}}\left( s\right) = \frac{1}... | A diagram of the dynamical complex dimensions of the golden flow is given in Figures 2.12 and 3.9. By Theorem 3.30 and the comment preceding it, we even know that the complex dimensions of the corresponding self-similar string are all simple, hence the dynamical and geometric complex dimensions correspond exactly. | No |
Theorem 7.28. Let \( {\mathcal{F}}_{\mathfrak{w}} \) be a self-similar flow of dimension \( D \) and with scaling ratios \( 1 > {r}_{1} \geq \cdots \geq {r}_{N} > 0 \) . Then \( s = D \) is the only complex dimension of \( {\mathcal{F}}_{\mathfrak{w}} \) on the real line. All complex dimensions are simple, and the resi... | In the lattice case, \( {\zeta }_{\mathfrak{w}}\left( s\right) \) is a rational function of \( {e}^{-{ws}} \), where \( w \) is the generator of \( {\mathcal{F}}_{\mathfrak{w}} \) . So, as a function of \( s \), it is periodic with period \( {2\pi i}/w \) . The complex dimensions \( \omega \) are obtained as the comple... | Yes |
Theorem 7.30 (The Prime Orbit Theorem with Error Term). Let \( {\mathcal{F}}_{\mathfrak{w}} \) be a languid suspended flow (i.e., \( {\zeta }_{\mathfrak{w}}^{\prime }/{\zeta }_{\mathfrak{w}} \) satisfies conditions \( \mathbf{{L1}} \) and \( \mathbf{{L2}} \) of Definition 5.2). Then we have the following equality betwe... | Proof. The first part of the theorem follows from the distributional explicit formula with error term (Theorem 5.18) and from the first part of Theorem 5.30, while the second part follows from the second part of Theorem 5.30. See also Theorem 5.17, in case there is no screen such that only \( D \) is visible. | Yes |
Corollary 7.36. Assume that the partial quotients \( {a}_{0},{a}_{1},\ldots \) of \( \alpha \) are bounded by b. Put \( B = {\pi }^{4}{e}^{-\left( {{w}_{1} + {w}_{2}}\right) D}/\left( {2{f}^{\prime }{\left( D\right) }^{3}}\right) \) (see also Equation (3.43) of Section 3.6). Then \( {\mathcal{F}}_{\mathfrak{w}} \) has ... | Proof. This follows from Theorem 3.34, if we note that for \( t = {2\pi }{q}_{k}/{w}_{1} \) , we have \( {q}_{k + 1}^{\prime } = {\alpha }_{k + 1}{q}_{k}^{\prime } \leq {2b}\left( {q}_{k}\right) {q}_{k}^{\prime } \leq {4b}\left( {q}_{k}\right) {q}_{k} \) . So the complex dimension close to \( D + {it} \) is located at ... | Yes |
Corollary 7.40. The best dimension-free region that \( {\mathcal{F}}_{\mathfrak{w}} \) can have is of size\n\n\[ \left\{ {\sigma + {it} : \sigma \geq D - O\left( {t}^{-2/\left( {M - 1}\right) }\right) }\right\} .\n\]\n\n(7.48)\n\nThe implied constant is positive and depends only on \( {w}_{1},\ldots ,{w}_{M} \) . | Let \( {w}_{1},\ldots ,{w}_{M} \) be b-approximable, where \( b : \lbrack 1,\infty ) \rightarrow {\mathbb{R}}^{ + } \) is an increasing function such that for every integer \( q \geq 1 \), \n\n\[ \left| {q{w}_{j} - {p}_{j}{w}_{1}}\right| \geq \frac{{w}_{1}}{b\left( q\right) }{q}^{-1/\left( {M - 1}\right) } \]\n\nfor \(... | Yes |
Theorem 8.1 (The distributional tube formula). Let \( \mathcal{L} \) be languid for some real exponent \( \kappa \) and a screen that does not pass through 0 . Then the volume of the (one-sided) tubular neighborhood of radius \( \varepsilon \) of the boundary of \( \mathcal{L} \) is given by the following distributiona... | Proof. Let \( \varphi \left( \varepsilon \right) \) be a smooth, compactly supported test function on \( \left( {0,\infty }\right) \) ; i.e., \( \varphi \in \mathbf{D}\left( {0,\infty }\right) \) . Then, by (8.3),\n\n\[ {\int }_{0}^{\infty }\varphi \left( \varepsilon \right) {v}_{\varepsilon }\left( x\right) {d\varepsi... | Yes |
If in Theorem 8.1, we assume in addition that all the visible complex dimensions of \( \mathcal{L} \) are simple, then the distributional tube formula (8.4) becomes\n\n\[ V\left( \varepsilon \right) = \mathop{\sum }\limits_{{\omega \in {\mathcal{D}}_{\mathcal{L}}\left( W\right) \smallsetminus \{ 0\} }}\operatorname{res... | Moreover if \( \mathcal{L} \) is assumed to be strongly languid rather than languid, then we may take \( W = \mathbb{C} \), and the error term \( \mathcal{R}\left( \varepsilon \right) \) vanishes identically. As in the statement of the second part of Theorem 8.1, this formula applies to test functions supported on a co... | Yes |
Theorem 8.7 (The pointwise tube formula). Let \( \mathcal{L} \) be a languid fractal string of dimension \( D < 1 \) and with exponent \( \kappa < 1 \) in hypotheses \( \mathbf{L}\mathbf{1} \) and \( \mathbf{{L2}} \), for a screen that does not pass through 0 . Then, for every \( \varepsilon > 0 \) , the volume of the ... | Proof. Let the fractal string \( \mathcal{L} \) be given by \( \mathcal{L} = {\left\{ {l}_{j}\right\} }_{j = 1}^{\infty } \) . Then, by (8.16), we have that \( {H}^{\left\lbrack 1\right\rbrack }\left( {{l}_{j} - {2\varepsilon }}\right) \) equals 1 for \( {l}_{j} > {2\varepsilon } \) and \( 1/2 \) for \( {l}_{j} = {2\va... | Yes |
If in Theorem 8.7, we assume in addition that all the visible complex dimensions of \( \mathcal{L} \) are simple, then the sum over the complex dimensions in the pointwise tube formula (8.17) becomes | \[ V\left( \varepsilon \right) = \mathop{\sum }\limits_{{\omega \in {\mathcal{D}}_{\mathcal{L}}\left( W\right) \smallsetminus \{ 0\} }}\operatorname{res}\left( {{\zeta }_{\mathcal{L}}\left( s\right) ;\omega }\right) \frac{{\left( 2\varepsilon \right) }^{1 - \omega }}{\omega \left( {1 - \omega }\right) } + \{ {2\varepsi... | Yes |
Lemma 8.13. Let \( \mathcal{L} \) be a generalized fractal string, given by a (local) positive measure \( \eta \) . If the pole of \( {\zeta }_{\mathcal{L}} \) at \( D \) is of order \( m \geq 1 \), then any pole at \( D + {it}\left( {\text{with}t \in \mathbb{R}}\right) \) is of order at most \( m \) . | Proof. Let \( \operatorname{Re}s = \sigma > D \) . Since \( \left| {{\zeta }_{\mathcal{L}}\left( s\right) }\right| \leq {\zeta }_{\mathcal{L}}\left( \sigma \right) \), we deduce that the function \( {\left( \sigma - D\right) }^{m}{\zeta }_{\mathcal{L}}\left( s\right) \) is bounded as \( \sigma \rightarrow {D}^{ + } \) ... | No |
Theorem 8.15 (Criterion for Minkowski measurability). Let \( \mathcal{L} \) be an ordinary fractal string that is languid for a screen passing between the vertical line \( \operatorname{Re}s = D \) and all the complex dimensions of \( \mathcal{L} \) with real part strictly less than \( D \), and not passing through 0 .... | Proof. Assume (i) and choose a screen such that only \( D \) is visible. By Theorem 5.18, the distributional explicit formula with error term, applied to \( \mathcal{L} \), and using Theorems 5.30 and 5.31, we obtain for \( {\mathcal{P}}_{\eta }^{\left\lbrack 1\right\rbrack }\left( x\right) = {N}_{\mathcal{L}}\left( x\... | Yes |
The Cantor string is a special case of the generalized Cantor string, with parameters \( a = 3 \) and \( b = 2 \) . This lattice string was studied in Chapter 1 and in Section 2.3.1. Note that the first length of the Cantor string equals \( \frac{1}{3} \) in Chapter 1 (so that the Cantor string fits in the unit interva... | In view of (8.32) and the discussion in Section 1.1.2, we have\n\n\[ V\left( \varepsilon \right) = \frac{1}{\log 3}{\left( 2\varepsilon \right) }^{1 - D}G\left( {{\log }_{3}{\left( 2\varepsilon \right) }^{-1}}\right) - {2\varepsilon }, \]\n\n(8.36)\n\nwhere \( D = {\log }_{3}2, r = 1/3 \), and \( G \) is the (nonconsta... | Yes |
Theorem 8.23. A lattice string is never Minkowski measurable and always has multiplicatively periodic oscillations of order \( D \), its dimension, in its geometry. | Proof (in the case of a single gap). Let \( \mathcal{L} \) be a self-similar lattice string with a single gap, normalized as in Remark 2.6. \( {}^{10} \) Choose a number \( \Theta \) with \( 0 \leq \Theta < D \) such that the first line of complex dimensions to the left of \( D \) lies to the left of the line \( \opera... | No |
Theorem 8.25 (Lattice strings with multiple gaps). Let \( \mathcal{L} \) be a lattice self-similar string with multiplicative generator \( r \) . Assume that the complex \( {di} \) - mensions of \( \mathcal{L} \) are all simple. Then for all \( \varepsilon \) with \( 0 < \varepsilon < \frac{1}{2}L{g}_{K}{r}_{N}^{-1} \)... | Proof. The theorem follows from the second part of Theorem 8.7 (the pointwise tube formula with error term). In light of the comment at the beginning of Section 6.4 (just above Remark 6.11), according to which \( \mathcal{L} \) is strongly languid with \( A = {L}^{-1}{g}_{K}^{-1}{r}_{N} \) and \( \kappa = 0 \) (thus al... | Yes |
Corollary 8.27. Let \( \mathcal{L} \) be a lattice self-similar string, with scaling ratios \( {r}_{1},\ldots ,{r}_{N}\left( {N \geq 2}\right) \) and gaps \( {g}_{1},\ldots ,{g}_{K}\left( {K \geq 1}\right) \), as in Section 2.1. Let \( \operatorname{Re}s = \Theta \) be the rightmost vertical line to the left of \( \ope... | Proof. We deduce formula (8.51)-with \( G = {G}_{1} \) and the coefficients of \( G \) given by (8.52) and (8.53) from the first part of the pointwise formula for the volume of the tubular neighborhoods, Theorem 8.7. To obtain the error estimate as given, we choose for the screen a vertical line \( \operatorname{Re}s =... | Yes |
Theorem 8.30. Let \( \mathcal{L} \) be a lattice self-similar string of total length \( L \) , with scaling ratios \( {r}_{1} = {r}^{{k}_{1}},\ldots ,{r}_{N} = {r}^{{k}_{N}} \) and gaps \( {g}_{1},\ldots ,{g}_{K} \) . Then the average Minkowski content of \( \mathcal{L} \) exists and is given by the finite positive num... | Proof. The existence and the computation of the average Minkowski content results from an application of Corollary 8.27. More precisely, according to (8.51) and (8.54), we have for all \( 0 \leq \varepsilon \leq 1 \) and some \( \Theta < D \) ,\n\n\[ \n{\varepsilon }^{-\left( {1 - D}\right) }V\left( \varepsilon \right)... | Yes |
Example 8.32 (The Fibonacci string). Recall from Section 2.3.2 that this is a lattice string with two lines of complex dimensions (see Figure 2.5 on page 43). Since all these complex dimensions are simple, the tube formula (8.40) gives | \[ V\left( \varepsilon \right) = \frac{2 + \phi }{5\log 2}\mathop{\sum }\limits_{{n \in \mathbb{Z}}}\frac{{\left( 2\varepsilon \right) }^{1 - D - {in}\mathbf{p}}}{\left( {D + {in}\mathbf{p}}\right) \left( {1 - D - {in}\mathbf{p}}\right) } - {2\varepsilon } + \frac{3 - \phi }{5\log 2}\mathop{\sum }\limits_{{n \in \mathb... | Yes |
Example 8.34 (A lattice string with multiple poles). An example of a lattice string \( \mathcal{L} \) with multiple poles was considered in Section 2.3.4. Recall that \( \mathcal{L} \) has one (discrete) line of complex dimensions above \( D \) , namely, \( \omega = D + {in}\mathbf{p} \) (with \( n \in \mathbb{Z}, D = ... | \[ V\left( \varepsilon \right) = \frac{4}{9\log 3}\mathop{\sum }\limits_{{n \in \mathbb{Z}}}\frac{{\left( 2\varepsilon \right) }^{1 - D - {in}\mathbf{p}}}{\left( {D + {in}\mathbf{p}}\right) \left( {1 - D - {in}\mathbf{p}}\right) } - \frac{\varepsilon }{2} \] \[ + \frac{1}{9{\left( \log 3\right) }^{2}}\mathop{\sum }\lim... | Yes |
Theorem 8.37. The volume \( V\left( \varepsilon \right) \) of the tubular neighborhoods of the non-lattice string \( \mathcal{L} \) is estimated by\n\n\[ V\left( \varepsilon \right) = \mathcal{M}{\varepsilon }^{1 - D} + O\left( {{\varepsilon }^{1 - D}{\left( \frac{\log \log {\varepsilon }^{-1}}{\log {\varepsilon }^{-1}... | Proof. The proof is similar to that of Theorems 7.37 and 7.41 on page 228. Note that for Theorem 8.37, the Tauberian argument of the last step of that proof is not needed, because Theorem 8.7 already gives \( V\left( \varepsilon \right) \) as a pointwise formula. Consequently, the exponent in the error term is best pos... | No |
Corollary 9.6. The inverse spectral problem (S) is not true in the mid-fractal case (i.e., when \( D = 1/2 \), see Figure 9.1). | On the other hand, it is true for every \( D \in \left( {0,1}\right), D \neq 1/2 \), if and only if the Riemann hypothesis holds. In the terminology of Section 6.3.2, the spectral operator is invertible for all fractal strings of dimension \( D \neq 1/2 \) if and only if the Riemann hypothesis holds. In that case, the ... | No |
The frequencies of the fundamental domain of the lattice \( {\mathbb{Z}}^{m} \) in \( {\mathbb{R}}^{m}\left( {m \in {\mathbb{N}}^{ * }}\right) \), with identification of opposite sides, are described by the classical Epstein zeta functions | \[ {\zeta }_{m}\left( s\right) = \mathop{\sum }\limits_{{v \in {\mathbb{Z}}^{m}\smallsetminus \{ 0\} }}\parallel v{\parallel }^{-s} \] where \( \parallel v\parallel \) is the Euclidean norm. These functions satisfy a functional equation, relating \( {\zeta }_{m}\left( s\right) \) and \( {\zeta }_{m}\left( {m - s}\right... | Yes |
Theorem 10.7. Let \( x \) be a positive real number and let\n\n\[ \nx = \mathop{\sum }\limits_{{k \in \mathbb{Z}}}{x}_{k}{a}^{k}\n\]\n\nbe the expansion of \( x \) in base a. Then the spectral counting function, \( {N}_{\nu }\left( x\right) \) , of the integral Cantor string with parameters \( a \) and \( b \) is given... | Proof. Observe that the digits of \( x \) form a finite sequence to the left in the sense that \( {x}_{k} = 0 \) for \( k \ll 0 \) . The expansion of \( x \) in base \( a \) allows us to obtain an expression for the integer part of \( {a}^{-n}x \) :\n\n\[ \n\left\lbrack {{a}^{-n}x}\right\rbrack = \left\lbrack {\mathop{... | Yes |
Corollary 10.8. The counting function of the frequencies, \( {N}_{\nu }\left( x\right) \), of an integral Cantor string jumps by\n\n\[ \n\frac{{b}^{n + 1} - 1}{b - 1} \n\]\n\n at integral values of \( x \) that are divisible by \( {a}^{n} \) and not by \( {a}^{n + 1}\left( {n \in \mathbb{N}}\right) \) . | Proof. Let \( x \) be exactly divisible by \( {a}^{n} \) . Then, there are two ways to represent \( x \) in base \( a \) :\n\n\[ \n{x}_{ + } = \mathop{\sum }\limits_{{k = n}}^{\infty }{x}_{k}{a}^{k}\;\text{ and }\;{x}_{ - } = \mathop{\sum }\limits_{{k = n + 1}}^{\infty }{x}_{k}{a}^{k} + \left( {{x}_{n} - 1}\right) {a}^... | Yes |
Theorem 10.9. At \( x = {a}^{n}\left( {n \in \mathbb{N}}\right) \), the spectral counting function \( {N}_{\nu }\left( x\right) \) jumps by at least \( {b}^{n} = {x}^{D} \) . Hence \( {N}_{\nu }\left( x\right) \) has jumps of order \( D \) at such points. | Proof of Theorem 10.9. The frequencies are the numbers \( k \cdot {a}^{j} \), with multiplicity \( {b}^{j} \) (with \( k \in {\mathbb{N}}^{ * }, j \in \mathbb{N} \) ). Thus the frequency \( {a}^{n} \) is found for all possible choices of \( k \) and \( j \) such that \( k{a}^{j} = {a}^{n} \) . We find \( j = n, n - m,\... | Yes |
Proposition 10.13. The function \( {K}_{\Lambda } \) has most of its mass concentrated around the integers:\n\n(i) \( {K}_{\Lambda }\left( x\right) \geq 0 \) for all \( x \in \mathbb{R} \),\n\n(ii) the total mass \( {\int }_{0}^{1}{K}_{\Lambda }\left( x\right) {dx} \) equals 1,\n\n(iii) \( {\int }_{0}^{1/\Lambda }{K}_{... | Proof. It is well known that\n\n\[ \n{K}_{\Lambda }\left( x\right) = \frac{1}{\Lambda }{\left( \frac{\sin {\pi \Lambda x}}{\sin {\pi x}}\right) }^{2}\n\]\n\nsee, e.g., [Fol, Section 8.5 and Exercise 33, p. 269]. This proves (i).\n\nProperty (ii) is clear since the constant coefficient of the Fourier series of \( {K}_{\... | Yes |
Lemma 10.14. For \( d \in \left( {0,1}\right) \) we have the estimate\n\n\[ \mathop{\sum }\limits_{{\left| n\right| < \Lambda }}\left( {1 - \frac{\left| n\right| }{\Lambda }}\right) \frac{1}{d + {in}\mathbf{p}} < \frac{1}{d}{e}^{{2\zeta }\left( 2\right) /{\mathbf{p}}^{2}}. \] | Proof. We combine the terms for positive and negative \( n \) to estimate\n\n\[ \left( {1 - \frac{\left| n\right| }{\Lambda }}\right) \left( {\frac{1}{d - {in}\mathbf{p}} + \frac{1}{d + {in}\mathbf{p}}}\right) < \frac{2d}{{d}^{2} + {n}^{2}{\mathbf{p}}^{2}} < \frac{2}{{n}^{2}{\mathbf{p}}^{2}}. \]\n\nHence their sum is b... | Yes |
Theorem 10.16. The function \( {N}_{\nu }\left( x\right) = {N}_{\nu, T}\left( x\right) \) is given by the following explicit formula: | \[ {N}_{\nu }\left( x\right) = {\operatorname{vol}}_{1}\left( T\right) x + \frac{1}{\log a}\mathop{\sum }\limits_{{\left| n\right| < \Lambda }}{c}_{n}\frac{{x}^{D + {in}\mathbf{p}}}{D + {in}\mathbf{p}}\zeta \left( {D + {in}\mathbf{p}}\right) + {\int }_{0}^{\infty }\left\{ {x{a}^{t}}\right\} {a}^{-{Dt}}{K}_{\Lambda }\le... | Yes |
Theorem 11.1. Let \( 0 < D < 1 \) and \( \mathbf{p} > 0 \) be given. Then there exists an integer \( n \neq 0 \) such that \( \zeta \left( {D + {in}\mathbf{p}}\right) \neq 0 \) . That is, the Riemann zeta function does not have an infinite sequence of critical zeros forming an arithmetic progression. | Proof of Theorem 11.1. Assume that \( \zeta \left( {D + {in}\mathbf{p}}\right) = 0 \) for all \( n \neq 0 \) . Let \( a = {e}^{{2\pi }/\mathbf{p}} \) and \( b = {a}^{D} \) . The generalized Cantor string \( {\mathcal{L}}_{D,\mathbf{p}} \) with these parameters has complex dimensions at all the points \( D + {in}\mathbf... | Yes |
Theorem 11.5. Let \( \mathbf{p} > 0 \) and \( D \in \left( {0,1}\right) \) be real numbers and let \( \Lambda \geq 2 \) . Suppose that \( \zeta \left( {D + {in}\mathbf{p}}\right) = 0 \) for all integers \( n \) such that \( 0 < \left| n\right| < \Lambda \) . Then \[ \Lambda < {60}\log \mathbf{p}{\left( \frac{\mathbf{p}... | In light of the functional equation satisfied by \( \zeta \left( s\right) \), we can assume without loss of generality that \( D \geq 1/2 \) . Thus the length of an arithmetic progression is bounded by \( O\left( \mathbf{p}\right) \) for \( D = 1/2 \), and by \( o\left( \mathbf{p}\right) \) for \( D > 1/2 \) . To estab... | No |
Lemma 11.6. Let \( m \in \mathbb{N} \) and \( \varepsilon = 1/\Lambda \) . Then\n\n\[ \n{N}_{\nu }\left( {a}^{m + \varepsilon }\right) - {N}_{\nu }\left( {a}^{m}\right) > {C}_{1}{a}^{mD},\n\]\n\nwhere \( {C}_{1} = {\int }_{0}^{1}{\left( \frac{\sin {\pi t}}{\pi t}\right) }^{2}{dt} > 9/{20} \) is the absolute constant of... | Proof. As in Section 10.3, let \( {N}_{T}\left( x\right) \) denote the geometric counting function of the truncated Cantor string, given by formula (10.14), or equivalently, by (10.15). Since \( {N}_{\nu }\left( x\right) = \mathop{\sum }\limits_{{\mu = 1}}^{\infty }{N}_{T}\left( {x/\mu }\right) \), and \( {N}_{T}\left(... | Yes |
Assume that \( \zeta \left( {D + {in}\mathbf{p}}\right) = 0 \) for \( 0 < \left| n\right| \leq \Lambda - 1 \) . Then, with \( \varepsilon = 1/\Lambda \) and for all integers \( m \geq 0 \), we have\n\n\[ \frac{{N}_{\nu }\left( {a}^{m + \varepsilon }\right) - {N}_{\nu }\left( {a}^{m}\right) }{\varepsilon } < {C}_{2}^{1/... | Proof. By Theorem 10.16 we have\n\n\[ {N}_{\nu }\left( {a}^{m + \varepsilon }\right) - {N}_{\nu }\left( {a}^{m}\right) = {a}^{m}\left( {{a}^{\varepsilon } - 1}\right) {\operatorname{vol}}_{1}\left( T\right) + \frac{{a}^{mD}}{\log a}\frac{{a}^{\varepsilon D} - 1}{D}\zeta \left( D\right) \]\n\n\[ + {\int }_{0}^{\infty }\... | Yes |
Lemma 11.8. If \( \zeta \left( {D + i\mathbf{p}}\right) = 0 \), then \( \frac{1}{1 - D} < {18}\log \mathbf{p} \) . | We obtain that\n\n\[ \n{C}_{1}\Lambda < \frac{1}{D} + \left( {\zeta \left( D\right) - \frac{1}{D - 1}}\right) + \frac{{18}\left( {{e}^{4/\mathbf{p}} - 1}\right) \log \mathbf{p}}{D}.\n\]\n\nNow \( \zeta \left( D\right) - \frac{1}{D - 1} \) is a bounded function for \( 1/2 \leq D < 1 \) . In fact, the function on the rig... | No |
Lemma 11.9. For \( \mathbf{p} > {10},{000} \), we have \( 1 - {aD} > \left( {1 - D}\right) {e}^{-{120}\left( {\log \mathbf{p}}\right) /\mathbf{p}} \) . | Proof. Use \( a = {e}^{{2\pi }/\mathbf{p}} \) and \( {e}^{x} - 1 < \frac{x}{1 - x} \) to estimate\n\n\[ \frac{1 - {aD}}{1 - D} = 1 - \left( {a - 1}\right) \frac{D}{1 - D} > 1 - \frac{2\pi D}{\left( {\mathbf{p} - {2\pi }}\right) \left( {1 - D}\right) }.\]\n\nEstimating \( D \) by 1 and \( 1/\left( {1 - D}\right) \) by L... | Yes |
Theorem 11.12. Let \( {\zeta }_{B} \) be a zeta function satisfying hypothesis \( \left( P\right) \) above. Let \( \mathbf{p} > 0 \) be arbitrary and let \( D \in W \cap \mathbb{R} \) be such that the vertical line \( \operatorname{Re}s = D \) lies entirely within \( W \) . Then there exists an integer \( n \neq 0 \) s... | Proof of Theorem 11.12. Assume that \( {\zeta }_{B}\left( {D + {in}\mathbf{p}}\right) = 0 \) for all \( n \neq 0 \) . As above, let \( \mathcal{L} = {\mathcal{L}}_{D,\mathbf{p}} \) be the generalized Cantor string with \( a = {e}^{{2\pi }/\mathbf{p}} \) and \( b = {a}^{D} \) . Then the generalized Cantor spray of \( \m... | Yes |
Theorem 11.14. Let \( \delta > 0 \) and assume that \( \rho < 1 \), where \( \rho \) is the exponent of Equation (11.16). Then, for infinitely many values of \( T \), tending to infinity, the set \[ \left\{ {n \in \mathbb{Z} : \left| n\right| \leq T,\left| {{\zeta }_{B}\left( {D + {in}\mathbf{p}}\right) }\right| \neq 0... | Proof. Suppose that the set defined by (11.17) contains fewer than \( {T}^{1 - \rho - \delta } \) elements, for all sufficiently large \( T \) . Let \( 0 < {n}_{1} < {n}_{2} < {n}_{3} < \ldots \) be the sequence of positive elements of the set (11.17) for \( T = \infty \) . Then we have that \( {n}_{j} \geq {j}^{1/\lef... | Yes |
Theorem 11.16. Let \( {\zeta }_{B} \) satisfy hypothesis \( \left( P\right) \) above. Then there do not exist \( D \in W \cap \mathbb{R} \) and \( \mathbf{p} > 0 \) such that \( {\zeta }_{B}\left( {D + {in}\mathbf{p}}\right) \rightarrow 0 \) as \( \left| n\right| \rightarrow \infty {.}^{5} \) | Proof. By Theorems 5.18 and 5.30 applied at level \( k = 0 \), we obtain the analogue of formula (11.12):\n\n\[ \nu = {W}_{B,\mathcal{L}}^{\left\lbrack 0\right\rbrack }\left( x\right) {dx} + \frac{1}{\log a}\mathop{\sum }\limits_{{n \in \mathbb{Z}}}{x}^{D - 1 + {in}\mathbf{p}}{\zeta }_{B}\left( {D + {in}\mathbf{p}}\rig... | Yes |
Theorem 11.17. Let \( K \) be a number field and \( {\chi }_{0} \) a character of a generalized ideal class group of \( K \) . Then the associated Hecke L-series, \( L\left( {s,{\chi }_{0}}\right) \) , has no infinite sequence of critical zeros forming an arithmetic progression. | Proof. Let \( L \) be the class field associated with this ideal class group, and let \( {\zeta }_{L} \) be the Dedekind zeta function of \( L \) . This is the Hecke \( L \) -series associated with the trivial character. Let \( \chi \) run over the characters of the ideal class group. According to a well-known result f... | Yes |
Theorem 11.19. Let\n\n\[ L\\left( {s,\\chi }\\right) = \\mathop{\\sum }\\limits_{{\\mu = 1}}^{\\infty }\\chi \\left( \\mu \\right) {\\mu }^{-s} \]\n\nbe a Dirichlet L-series with character \( \\chi \) . Let \( \\omega \) be a complex number with real part \( d \\geq 1/2 \), and let \( p > 0 \) . Then \( {}^{8}L\\left( ... | The argument is based on the idea of studying the frequencies of the \ | No |
Lemma 11.20. Let \( N, M \geq 0,0 < r < 1 \) and \( a > 1 \) . Then there exists an integer \( \widetilde{n} \) satisfying\n\n\[ N < \widetilde{n} < N + \left( {1 - {\log }_{a}r}\right) M + 1, \]\n\n(11.22)\n\nsuch that if \( \begin{Vmatrix}{a}^{\mu }\end{Vmatrix} \leq {a}^{\mu - \widetilde{n}} \) then \( \begin{Vmatri... | Proof. Order the numbers \( \mu - {\log }_{a}\begin{Vmatrix}{a}^{\mu }\end{Vmatrix} \) that are greater than \( N \) in increasing order, together with \( N \), to obtain a sequence of values\n\n\[ N = {v}_{0} < {v}_{1} < {v}_{2} < \cdots < \infty . \]\n\nFor \( \mu = 0 \), and in general if \( {a}^{\mu } \) is an inte... | Yes |
Let the numbers \( N, M \geq 0,0 < r < 1 \) and \( a > 1 \) be such that\n\n\[ a\left( {1 - r{a}^{-N}}\right) \geq 1 \]\n\nand\n\n\[ {a}^{{M}^{2}}\left( {{\left( 1 + r{a}^{-N}\right) }^{M} - {\left( 1 - r{a}^{-N}\right) }^{M}}\right) \leq 1. \]\n\nLet \( \widetilde{n} \) be as in Lemma 11.20. If there do not exist inte... | Proof. By Lemma 11.20, \( \alpha = {a}^{m}\left( {1 + \theta {a}^{-\widetilde{n}}}\right) \) for some real number \( \theta \) such that \( \left| \theta \right| \leq r \) . If \( \alpha = 1 \), then \( {a}^{m}\left( {1 - r{a}^{-\widetilde{n}}}\right) \leq 1 \) and hence \( a\left( {1 - r{a}^{-N}}\right) < 1 \) . But t... | Yes |
Lemma 11.22. Let the numbers \( a, N, M, r \) and the integers \( \widetilde{n}, m,\alpha \) be as in Lemmas 11.20 and 11.21. Assume that\n\n\[ \n{a}^{N}\log a \geq 1 \n\]\n\n(11.28)\n\n\[ \nN - M \geq {\log }_{a}\left( {3/2}\right) \n\]\n\n(11.29)\n\n\[ \nr \leq 1/2\text{.} \n\]\n\n(11.30)\n\nLet \( w \in \mathbb{N} \... | Proof. If \( l \geq n + 1 \), then\n\n\[ \nk{a}^{l} - {a}^{n} \geq {a}^{\widetilde{n} + w}\left( {a - 1}\right) > {a}^{w}{a}^{N}\left( {a - 1}\right) > {a}^{w}{a}^{N}\log a. \n\]\n\nHence by (11.28), if \( \left| {k{a}^{l} - {a}^{n}}\right| \leq {a}^{w} \), then \( l \leq n \) . Write \( \mu = n - l \), so that \( 0 \l... | Yes |
Example 11.24 (The projective line). If \( C = {\mathbb{P}}^{1} \), that is, if the curve has genus \( g = 0 \), then \( R = {\mathbb{F}}_{q}\left\lbrack X\right\rbrack \), the ring of polynomials over \( {\mathbb{F}}_{q} \) . In this case, every ideal of \( R \) is generated by a single polynomial. The number of ideal... | \[ \mathop{\sum }\limits_{\mathfrak{a}}{q}^{-s\deg \mathfrak{a}} = \mathop{\sum }\limits_{{n = 0}}^{\infty }{q}^{n}{q}^{-{ns}} = \frac{1}{1 - {q}^{1 - s}}. \] There is only one point at infinity, and the corresponding factor in the Euler product is \[ \frac{1}{1 - {q}^{-s}}. \] Thus, the zeta function of \( {\mathbb{P}... | Yes |
The curve given by \( {y}^{2} = {x}^{3} - x \) over \( {\mathbb{F}}_{3} \) has genus 1 . Its zeta function is | \[ \frac{1 + {3}^{1 - {2s}}}{\left( {1 - {3}^{-s}}\right) \left( {1 - {3}^{1 - s}}\right) } = 1 + 4\mathop{\sum }\limits_{{n = 1}}^{\infty }{3}^{-{ns}}\frac{{3}^{n} - 1}{2}, \] | Yes |
The Klein curve is given by the equation\n\n\[ \n{x}^{3}y + {y}^{3}z + {z}^{3}x = 0.\n\]\n\nOver \( {\mathbb{F}}_{2} \) it has genus 3, with zeta function given by\n\n\[ \n{2}^{2s}\frac{1 + 5 \cdot {2}^{-{3s}} + 8 \cdot {2}^{-{6s}}}{\left( {1 - {2}^{-s}}\right) \left( {1 - {2}^{1 - s}}\right) }\n\] | \[ \n= {2}^{2s} + 3 \cdot {2}^{s} + 7 + {20} \cdot {2}^{-s} + {46} \cdot {2}^{-{2s}} + {14}\mathop{\sum }\limits_{{n = 3}}^{\infty }{2}^{-{ns}}\left( {{2}^{n} - 1}\right) ,\n\]\n\nand the Weyl term associated with a string \( \mathcal{L} = {\left\{ {l}_{j}\right\} }_{j = 1}^{\infty } \) is\n\n\[ \nW\left( x\right) = {1... | Yes |
Consider the zeta function\n\n\[ \n{\zeta }_{X}\left( s\right) = \frac{1 - {5x} + 5{x}^{2}}{\left( {1 - x}\right) \left( {1 - {5x}}\right) },\;x = {5}^{-s}.\n\]\n\nAs a Dirichlet series,\n\n\[ \n{\zeta }_{X}\left( s\right) = 1 + \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{5}^{n} - 1}{4}{x}^{n},\;x = {5}^{-s}.\n\] | One checks that this function satisfies the functional equation (11.53). Moreover, the numerator factors as \( 1 - {5x} + 5{x}^{2} = \left( {1 - \left( {3 - \phi }\right) x}\right) \left( {1 - \left( {2 + \phi }\right) x}\right) \) , where \( \phi \) is the golden ratio. Since \( 3 - \phi < \sqrt{5} < 2 + \phi \), the ... | Yes |
Let \( \zeta \) be the Riemann zeta function, and let \( {L}_{1},\ldots ,{L}_{2n} \) be \( L \) -series associated with \( {2n} \) independent nontrivial real characters (see Appendix A, Section A.2). \( {}^{1} \) Then, for any choice of real numbers \( {c}_{1},\ldots ,{c}_{2n} \) , the function \[ \zeta \left( s\right... | By suitably choosing the coefficients, one can arrange for this function to have a sequence of zeros in an arithmetic pro- \( {}^{1} \) Note that by assumption, only \( \zeta \left( s\right) \) has a pole (at \( s = 1 \) ). gression of length \( n \) . Indeed, choosing any sequence \( D + {ik}\mathbf{p}, k = 1,\ldots, ... | No |
The function \[ \mathop{\prod }\limits_{{k = 1}}^{n}\zeta \left( {\frac{1}{2} + \frac{s}{k}}\right) \] shows that a function with \( n \) poles can have \( n \) zeros in arithmetic progression. | Indeed, let \( 1/2 + {i\gamma } \) be a critical zero of \( \zeta \left( s\right) \), with \( \gamma > 0 \) . Then the values \( s = {ki\gamma } \), for \( k = 1,\ldots, n \), are zeros of the above function in the upper half-plane. | Yes |
Let \( {\zeta }_{K}\left( s\right) \) be the zeta function of an algebraic number field \( K \) . As is well known, this function has zeros with real part \( 1/2 \) . Then the function \[ \zeta \left( {\frac{1}{2} + s}\right) + {c}_{1}{\zeta }_{K}\left( {\frac{1}{2} + {c}_{2}s}\right) \] with a suitable choice of the r... | Indeed, if \( {\zeta }_{K} \) does not have a double zero itself, then one can first choose \( {c}_{2} \) so that both functions have a common zero at a point \( s = {i\gamma } \) (here, \( \gamma \) is as in the previous example). One can then adjust \( {c}_{1} \) so that the derivative also vanishes at that point. | No |
Theorem 12.21 (Tube Formula). The volume of the inner \( \varepsilon \) -neighborhood of the von Koch snowflake curve \( \partial \Omega \) is given by the following pointwise formula: | \[ V\left( \varepsilon \right) = {G}_{1}\left( \varepsilon \right) {\varepsilon }^{2 - D} + {G}_{2}\left( \varepsilon \right) {\varepsilon }^{2}, \] where \( {G}_{1} \) and \( {G}_{2} \) are periodic functions (of multiplicative period 3) given by \[ {G}_{1}\left( \varepsilon \right) = \frac{1}{\log 3}\mathop{\sum }\li... | Yes |
Theorem 12.27. The tube formula of a self-similar tiling is given by\n\n\[ \n{V}_{\text{til }}\left( \varepsilon \right) = \mathop{\sum }\limits_{{\omega \in {\mathcal{D}}_{\text{til }}}}\operatorname{res}\left( {{\zeta }_{\text{til }}\left( {\varepsilon, s}\right) ;\omega }\right) ,\n\]\n\nwhere \( {\mathcal{D}}_{\tex... | In the special case when all the poles of \( {\zeta }_{\text{til }} \) are simple and the curvature matrix \( \mathbf{\kappa } \) is constant for \( \varepsilon < \min \left\{ {{g}_{1},\ldots ,{g}_{K}}\right\} \), the tube formula is of the form (12.13), with \( W = \mathbb{C} \) and \( \mathcal{R}\left( \varepsilon \r... | Yes |
Example 12.28 (The Koch Tiling). The tube formula for the Koch tiling KT is of the form\n\n\[ \n{V}_{\mathrm{{KT}}}\left( \varepsilon \right) = \frac{g}{\log 3}\mathop{\sum }\limits_{{n \in \mathbb{Z}}}{c}_{n}{\left( \frac{\varepsilon }{g}\right) }^{2 - D - {in}\mathbf{p}} + {3}^{3/2}{\varepsilon }^{2} + \frac{1}{1 - 2... | From (12.27), we clearly read off that\n\n\[ \n{\mathcal{D}}_{\mathfrak{s}} = \{ D + {in}\mathbf{p} : n \in \mathbb{Z}\} \;\text{ and }\;{\mathcal{D}}_{\mathrm{{KT}}} = {\mathcal{D}}_{\mathfrak{s}} \cup \{ 0,1\} . \]\n\n(12.28)\n\nIndeed, the elements of \( {\mathcal{D}}_{\mathfrak{s}} \) are precisely the poles of the... | Yes |
Theorem 12.30 ([HamLap, Theorem 4.5]). Almost surely, the random zeta function \( {\zeta }_{\mathcal{L}}\left( s\right) \) of the random self-similar string admits a meromorphic continuation to a nontrivial open half-plane \( \operatorname{Re}s > D - \tau \), where \( D \) is the Minkowski dimension of \( \mathcal{L} \... | One deduces from the above result (and our earlier work on explicit formulas, see Chapters 5 and 8) an inner tube formula for random self-similar strings (see Theorem 7.6 in [HamLap]). In particular, for almost every realization of \( \mathcal{L} \), the volume of the inner tubular neighborhoods is given, for \( \varep... | Yes |
Lemma 1.2.2. Homotopy is an equivalence relation. | Proof. Reflexive: \( \;f \) is homotopic to \( f \) via the homotopy of waiting (i.e., changing nothing) for one unit of time.\n\nSymmetric: If \( f \) is homotopic to \( g \), then \( g \) is homotopic to \( f \) via the homotopy of running the original homotopy backwards in time.\n\nTransitive: If \( f \) is homotopi... | Yes |
Example 1.2.7. Let us regard \( X : {\mathbb{R}}^{n} - \{ \left( {0,\ldots ,0}\right) \} \) as the space of nonzero vectors \( \left\{ {v \in {\mathbb{R}}^{n} \mid v \neq 0}\right\} \) . Then \( A = {S}^{n - 1} = \left\{ {v \in {\mathbb{R}}^{n} \mid \parallel v\parallel = 1}\right\} \) is a subspace of \( X \), and is ... | The map\n\n\[ F : X \times I \rightarrow A \]\n\n given by\n\n\[ F\left( {v, t}\right) = \parallel v{\parallel }^{t}\left( {v/\parallel v\parallel }\right) \]\n\ngives a homotopy rel \( A \) from the retraction \( {f}_{0}\left( v\right) = v/\parallel v\parallel \) to the identity map \( {f}_{1}\left( v\right) = v \) . ... | Yes |
Lemma 1.2.12. For any space \( X,{cX} \) is contractible. | Proof. Let \( F : {cX} \times I \rightarrow {cX} \) be defined by\n\n\[ F\left( {\left( {x, s}\right), t}\right) = \left( {x,\max \left( {s, t}\right) }\right) . \]\n | Yes |
Lemma 2.1.2. The fundamental group \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) is a group. | Proof. The identity element of this group is represented by the constant map \( i : {S}^{1} \rightarrow \) \( \left\{ {x}_{0}\right\} \) and the inverse of \( f : \left( {{S}^{1},1}\right) \rightarrow \left( {X,{x}_{0}}\right) \) is represented by the map \( g : \left( {{S}^{1},1}\right) \rightarrow \) \( \left( {X,{x}... | Yes |
Lemma 2.1.3. Let \( {x}_{0},{x}_{1} \in X \) . Choose a path \( \varphi \) from \( {x}_{0} \) to \( {x}_{1} \), i.e., a map \( \varphi : I \rightarrow X \) with \( \varphi \left( 0\right) = {x}_{0} \) and \( \varphi \left( 1\right) = {x}_{1} \) . Let \( \bar{\varphi } : I \rightarrow X \) be the map given by \( \bar{\v... | \[ g\left( t\right) = \left\{ \begin{array}{ll} \varphi \left( {3t}\right) & 0 \leq t \leq \frac{1}{3} \\ f\left( {{3t} - 1}\right) & \frac{1}{3} \leq t \leq \frac{2}{3} \\ \bar{\varphi }\left( {{3t} - 2}\right) & \frac{2}{3} \leq t \leq 1 \end{array}\right. \] | Yes |
Theorem 2.1.5. Let \( f : X \rightarrow Y \) with \( f\left( {x}_{0}\right) = {y}_{0} \) . If \( f \) is a homotopy equivalence, then \( {f}_{ * } : {\pi }_{1}\left( {X,{x}_{0}}\right) \rightarrow {\pi }_{1}\left( {Y,{y}_{0}}\right) \) is an isomorphism. | Proof. Suppose that \( g : Y \rightarrow X \) with \( g\left( {y}_{0}\right) = {x}_{0} \), that \( {gf} : X \rightarrow X \) is homotopic to the identity rel \( {x}_{0} \), and that \( {fg} : Y \rightarrow Y \) is homotopic to the identity rel \( {y}_{0} \) . Then the theorem is very easy to prove. But we are not makin... | No |
Corollary 2.1.6. Let \( X \) be a contractible space. Then \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) is the trivial group. | Proof. This is clearly true if \( X \) is the space consisting of the point \( {x}_{0} \) alone, as then every \( f : \left( {{S}^{1},1}\right) \rightarrow \left( {X,{x}_{0}}\right) \) is the constant map to the point \( {x}_{0} \) . Then it is also true for \( X \) contractible by Theorem 2.1.5. | Yes |
Let \( G = \mathbb{Z} \) act on \( Y = \mathbb{R} \) by \( n\left( r\right) = r + n, n \in G \) and \( r \in \mathbb{R} \) . Let \( X = G \smallsetminus Y \) . Then \( X \) is homeomorphic to \( {S}^{1} \), so we see that \( {\pi }_{1}\left( {{S}^{1},1}\right) = \mathbb{Z} \) . | Note that we may identify the covering projection \( p : Y \rightarrow X \) with the covering projection in Example 2.2.3(iia). It is worth being completely explicit here. Let \( \pi : \mathbb{R} \rightarrow {S}^{1} \) by \( \pi \left( t\right) = \exp \left( {2\pi it}\right) \) . Let \( d \) be an integer and let \( {\... | Yes |
Theorem 2.2.8. Let \( p : Y \rightarrow X \) be a covering projection and let \( {y}_{0} \in Y \) and \( {x}_{0} \in X \) be points with \( p\left( {y}_{0}\right) = {x}_{0} \). Let \( E \) be an arbitrary connected and locally path connected space and let \( {e}_{0} \) be a point in \( E \). Let \( f : \left( {E,{e}_{0... | (Since \( f = p\widetilde{f} \), the condition in the theorem is obviously necessary. The point of the theorem is that it is sufficient.) | Yes |
Corollary 2.2.21. Under Hypotheses 2.2.17:\n\nEvery \( X \) has a simply-connected cover \( p : \widetilde{X} \rightarrow X \), unique up to equivalence. \( \widetilde{X} \) is the universal cover of \( X \), and \( X \) is the quotient of \( \widetilde{X} \) by the group of covering translations. Also, if \( Y \) is a... | Proof. This is a direct consequence of Theorem 2.2.19, and our earlier results, taking \( H \) to be the trivial subgroup of \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) . | Yes |
Theorem 2.3.1. Let \( X = {X}_{1} \cup {X}_{2} \) and suppose that \( {X}_{1},{X}_{2} \), and \( A = {X}_{1} \cap {X}_{2} \) are all open, path connected subsets of \( X \) . Let \( {x}_{0} \in A \) . Then \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) is the free product with amalgamation | \[ {\pi }_{1}\left( {X,{x}_{0}}\right) = {\pi }_{1}\left( {{X}_{1},{x}_{0}}\right) { * }_{{\pi }_{1}\left( {A,{x}_{0}}\right) }{\pi }_{1}\left( {{X}_{2},{x}_{0}}\right) . \] In other words, if \( {i}_{1} : A \rightarrow {X}_{1} \) and \( {i}_{2} : A \rightarrow {X}_{2} \) are the inclusions, then \( {\pi }_{1}\left( {X... | Yes |
Corollary 2.3.3. For \( n > 1 \), the \( n \) -sphere \( {S}^{n} \) is simply connected. | Proof. We regard \( {S}^{n} \) as the unit sphere in \( {\mathbb{R}}^{n + 1} \) . Let \( {X}_{1} = {S}^{n} - \{ \left( {0,0,\ldots ,0,1}\right) \} \) and \( {X}_{2} = {S}^{n} - \{ \left( {0,0,\ldots ,0, - 1}\right) \} \) . Then \( {X}_{1} \) and \( {X}_{2} \) are both homeomorphic to \( {\mathring{D}}^{n} \), so are pa... | Yes |
Regard \( {S}^{n} \) as the unit sphere in \( {\mathbb{R}}^{n + 1} \) and let \( {\mathbb{Z}}_{2} \) act on \( {S}^{n} \), where the nontrivial element \( g \) of \( {\mathbb{Z}}_{2} \) acts via the antipodal map, \( g\left( {{z}_{1},\ldots ,{z}_{n + 1}}\right) = \left( {-{z}_{1},\ldots , - {z}_{n + 1}}\right) \). The ... | Note that \( p : {S}^{0} \rightarrow \mathbb{R}{P}^{0} \) is the map from the space of two points to the space of one point, and \( p : {S}^{1} \rightarrow \mathbb{R}{P}^{1} \) may be identified with the cover in Example 2.2.3(iib) for \( n = 2 \). But for \( n > 1 \), by Corollary 2.3.3 and Theorem 2.2.6 we see that \... | No |
Corollary 2.3.6. The fundamental group \( {\pi }_{1}\left( {{R}_{n},{r}_{0}}\right) \) is the free group on the \( n \) elements \( {\alpha }_{k} = {\left( {i}_{k}\right) }_{ * }\left( {g}_{k}\right) \), where \( {g}_{k} \) is a generator of \( {\pi }_{1}\left( {{\left( {S}^{1}\right) }_{k},{\left( 1\right) }_{k}}\righ... | Proof. We proceed by induction on \( n \) .\n\nFor \( n = 1 \) this is Example 2.2.7.\n\nNow suppose that \( n \geq 1 \) and that the theorem is true for \( n \) . Write \( {R}_{n + 1} = {X}_{1} \cup {X}_{2} \) where:\n\n\[ \n{X}_{1} = \mathop{\bigcup }\limits_{{k = 1}}^{n}{\left( {S}^{1}\right) }_{k} \cup \left\{ {{\l... | Yes |
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