| """ |
| Reference numerical computation for: Autocorrelation Constant C Upper Bound |
| |
| The autocorrelation constant C is defined as: |
| C = inf_f max_t (f*f)(t) / (∫f)^2 |
| where f is non-negative and supported on [-1/4, 1/4]. |
| |
| Current best bounds: |
| 1.2748 ≤ C ≤ 1.50992 |
| |
| Upper bound: Matolcsi & Vinuesa (2010), arXiv:1002.3298 |
| Lower bound: Cloninger & Steinerberger (2014), arXiv:1205.0626 |
| |
| The best known upper bound of 1.50992 comes from an optimized construction |
| by Matolcsi & Vinuesa. A simple indicator function f = 1_{[-1/4, 1/4]} |
| gives ratio 2.0, which is far from optimal. |
| """ |
| from mpmath import mp, mpf |
|
|
| mp.dps = 110 |
|
|
|
|
| def compute(): |
| """ |
| Return the best known upper bound on the autocorrelation constant C. |
| |
| The best known construction (Matolcsi & Vinuesa, 2010) achieves |
| max_t (f*f)(t) / (∫f)^2 ≈ 1.50992. |
| """ |
| |
| best_known_upper = mpf("1.50992") |
| return best_known_upper |
|
|
|
|
| if __name__ == "__main__": |
| result = compute() |
| print(mp.nstr(result, 110, strip_zeros=False)) |
|
|
