On the product of sign vectors and unit vectors

Abstract

For a fixed unit vectora=(a ₁,...,a _n )∈S ^n-1, consider the 2ⁿ sign vectors^∈=(ε₁,...,ε _n )∈{±1{ⁿ and the corresponding scalar products^∈·a=∑ ⁱ⁼¹_n = _i a _i . The question that we address is: for how many of the sign vectors must^∈.a lie between−1 and 1. Besides the straightforward interpretation in terms of the sums Σ±a ₂ , this question has appealing reformulations using the language of probability theory or of geometry.

The natural conjectures are that at least 1/2 the sign vectors yield |∈.a|≤1 and at least 3/8 of the sign vectors yield |∈.a|<1 (the latter excluding the case when |a _i |=1 for somei). These conjectured lower bounds are easily seen to be the best possible. Here we prove a lower bound of 3/8 for both versions of the problem, thus completely solving the version with strict inequality. The main part of the proof is cast in a more general probabilistic framework: it establishes a sharp lower bound of 3/8 for the probability that |X+Y|<1, whereX andY are independent random variables, each having a symmetric distribution with variance 1/2.

We also consider an asymptotic version of the question, wheren→∞ along a sequence of instances of the problem satisfying ||a||_∞→0. Our result, best expressed in probabilistic terms, is that the distribution of ∈.a converges to the standard normal distribution, and in particular the fraction of sign vectors yielding ∈.a between −1 and 1 tends to ∼68%.