An improved lower bound for the minimum-autocorrelation constant
Abstract
We construct nonnegative bounded compactly supported functions on the real line that give the lower bound πβ₯ 0.4103779774321214877852987β¦ for the minimum-autocorrelation constant. This improves the explicitly certified bound 2378625/5958277=0.39921356β¦ reported by Russell. The construction combines a point mass with a monotone degree-ten polynomial density; a one-sided atom-removal argument then produces ordinary functions. Its numerical content reduces to exact rational polynomial identities and 20 Bernstein subinterval certificates. The finite certificate, actual overlap integrals, admissibility, mass formula, and ordinary-function witness bounds have been formalized in Lean 4.19.0; the pinned proof sources and recorded audit are supplied. The passage to the supremum constant is given separately as a mathematical argument. As applications, the same construction yields an upper bound below $1.5610183$ for the large-multiplicity generalized-difference-basis coefficient and a feasible squared mass below $2.436778$ for a continuous sparse-ruler relaxation. An appendix quantifies the loss under independent sampling followed by completion. No improved bound for the complete sparse ruler constant, or global optimality of the construction, is asserted.
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BibTeX
@misc{chen2026improved,
title = {{An improved lower bound for the minimum-autocorrelation constant}},
author = {Shenai Chen and EnHao Gu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18888},
eprint = {2610.18888},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18888},
}
RIS
TY - UNPB
TI - An improved lower bound for the minimum-autocorrelation constant
AU - Shenai Chen
AU - EnHao Gu
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.18888
UR - https://arxiv.org/abs/2610.18888
AB - We construct nonnegative bounded compactly supported functions on the real line that give the lower bound πβ₯ 0.4103779774321214877852987β¦ for the minimum-autocorrelation constant. This improves the explicitly certified bound 2378625/5958277=0.39921356β¦ reported by Russell. The construction combines a point mass with a monotone degree-ten polynomial density; a one-sided atom-removal argument then produces ordinary functions. Its numerical content reduces to exact rational polynomial identities and 20 Bernstein subinterval certificates. The finite certificate, actual overlap integrals, admissibility, mass formula, and ordinary-function witness bounds have been formalized in Lean 4.19.0; the pinned proof sources and recorded audit are supplied. The passage to the supremum constant is given separately as a mathematical argument. As applications, the same construction yields an upper bound below $1.5610183$ for the large-multiplicity generalized-difference-basis coefficient and a feasible squared mass below $2.436778$ for a continuous sparse-ruler relaxation. An appendix quantifies the loss under independent sampling followed by completion. No improved bound for the complete sparse ruler constant, or global optimality of the construction, is asserted.
ER -