PreprintCC BY 4.0arXiv
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.
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BibTeX
@misc{giraudin2026bounds,
title = {{New bounds for the integer Chebyshev constant of [0,1]}},
author = {David Niedbala Giraudin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00386},
eprint = {2610.00386},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00386},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - New bounds for the integer Chebyshev constant of [0,1]
AU - David Niedbala Giraudin
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.00386
UR - https://arxiv.org/abs/2610.00386
AB - 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.
ER -