A relative anti-concentration inequality

TL;DR

This paper investigates a conjecture about the probability that a random vector's inner product with one of two vectors exceeds that with the other, relating it to their Euclidean norms and structural properties.

Contribution

The paper proposes a conjecture on anti-concentration inequalities for inner products and provides partial results supporting its validity.

Findings

Partial results support the conjecture

Probability bound relates to Euclidean norms

Accounts for arithmetic structure in vectors

Abstract

Given two vectors in Euclidean space, how unlikely is it that a random vector has a larger inner product with the shorter vector than with the longer one? When the random vector has independent, identically distributed components, we conjecture that this probability is no more than a constant multiple of the ratio of the Euclidean norms of the two given vectors, up to an additive term to allow for the possibility that the longer vector has more arithmetic structure. We give some partial results to support the basic conjecture.