Upper bounds for the constants of Bennett's inequality and the Gale--Berlekamp switching game

TL;DR

This paper establishes new upper bounds for Bennett's inequality constants using constructive methods and applies these results to improve bounds in the Gale--Berlekamp switching game, enhancing previous estimates from past decades.

Contribution

The paper provides a constructive proof that bounds the Bennett inequality constant by rac{8}{5} for specific p-parameter ranges and applies these bounds to improve estimates in the Gale--Berlekamp switching game.

Findings

Bound C_{p_1,p_2} b1 a9; b1 rac{8}{5} for p_1,p_2 b1 a9; p_1,p_2 b1 a9 [2,a0a9] or p_1=p_2=p b1 a9 [1,a0a9]

Constructive approach yields explicit bounds for constants in Bennett's inequality.

Improved bounds for the Gale--Berlekamp switching game constants, surpassing previous estimates by Brown-Spencer and Carlson-Stolarski.

Abstract

In $1977$, G. Bennett proved, by means of non-deterministic methods, an inequality which plays a fundamental role in a series of optimization problems. More precisely, Bennett's inequality shows that, for $p_{1},p_{2} \in\lbrack1,\infty]$ and all positive integers $n_{1},n_{2}$, there exists a bilinear form $A_{n_{1},n_{2}}\colon\left( \mathbb{R}^{n_{1}},\left\Vert \cdot\right\Vert _{p_{1}}\right) \times\left( \mathbb{R}^{n_{2}},\left\Vert \cdot\right\Vert _{p_{2}}\right) \longrightarrow\mathbb{R}$ with coefficients $\pm1$ satisfying \[ \left\Vert A_{n_{1},n_{2}}\right\Vert \leq C_{p_{1},p_{2}}\max\left\{ n_{1}^{1-\frac{1}{p_{1}}}n_{2}^{\max\left\{ \frac{1}{2}-\frac{1}{p_{2} },0\right\} },n_{2}^{1-\frac{1}{p_{2}}}n_{1}^{\max\left\{ \frac{1}{2} -\frac{1}{p_{1}},0\right\} }\right\} \] for a certain constant $C_{p_{1},p_{2}}$ depending just on $p_{1},p_{2}$; moreover, the exponents of $n_{1},n_{2}$ cannot be improved. In this paper, using a constructive approach, we prove that $C_{p_{1},p_{2}}\leq\sqrt{8/5}$ whenever $p_{1},p_{2}\in\left[ 2,\infty\right] $ or $p_{1}=p_{2}=p\in\left[ 1,\infty\right] $. Our techniques are applied to provide new upper bounds for the constants of a combinatorial game, known as Gale--Berlekamp switching game or unbalancing lights problem. As a consequence, we improve estimates obtained by Brown and Spencer in $1971$ and by Carlson and Stolarski in $2004$.