New bounds for the integer Chebyshev constant of [0,1]

Abstract

The integer Chebyshev constant of [0,1] has been bracketed by 0.4213 <= t_Z([0,1]) <= 0.422685 since the work of Pritsker (2005) and of Flammang (2014). We prove 0.4222286000 <= t_Z([0,1]) <= 0.4226846976, narrowing the interval from 1.39e-3 to 4.56e-4. Both bounds are established by explicit certificates. The upper bound is carried by an integer polynomial with 92 factors and explicit integer exponents; the lower bound by a discrete measure on 947 cells together with a library of 206 irreducible polynomials, through an inequality in which the terms paying for divisibility are made explicit. Two self-contained scripts, included as ancillary files, re-derive both bounds from the certificates alone.

arXiv comment7 pages. Ancillary files: two JSON certificates, two standalone verifiers (numpy, mpmath, sympy). v2: upper bound in the abstract and Corollary 3 corrected to 0.4226846976, the upward rounding of the bound in Theorem 1; the gain over [P05] in Section 1 is 9.3e-4, not 9.1e-4; reference to the independent work of Qu and Yang (arXiv:2609.39505) added. Theorems, certificates and files unchanged