Some explorations on two conjectures about Rademacher sequences

TL;DR

This paper investigates two conjectures about Rademacher sequences, providing new formulations for the first and proving the second conjecture for sequences of length up to seven.

Contribution

It introduces new equivalent formulations for the first conjecture and verifies the second conjecture for small sequence lengths.

Findings

New formulations for the first conjecture, including topological and combinatorial versions.

Proof that the second conjecture holds for sequences with length up to 7.

Abstract

In this paper, we explore two conjectures about Rademacher sequences. Let $(\epsilon_i)$ be a Rademacher sequence, i.e., a sequence of independent $\{-1,1\}$-valued symmetric random variables. Set $S_n=a_1\epsilon_1+\cdots+a_n\epsilon_n$ for $a=(a_1,\dots,a_n)\in \mathbb{R}^n$. The first conjecture says that $P\ (\ |S_n\ |\leq \|a\|\ )\geq\frac{1}{2}$ for all $a\in \mathbb{R}^n$ and $n\in \mathbb{N}$. The second conjecture says that $P\ (\ |S_n\ |\geq\|a\|\ )\geq \frac{7}{32}$ for all $a\in \mathbb{R}^n$ and $n\in \mathbb{N}$. Regarding the first conjecture, we present several new equivalent formulations. These include a topological view, a combinatorial version and a strengthened version of the conjecture. Regarding the second conjecture, we prove that it holds true when $n\leq 7$.