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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 bound from Matolcsi & Vinuesa (2010)
best_known_upper = mpf("1.50992")
return best_known_upper
if __name__ == "__main__":
result = compute()
print(mp.nstr(result, 110, strip_zeros=False))
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