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=1n = 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%.

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This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.

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Holzman, R., Kleitman, D.J. On the product of sign vectors and unit vectors. Combinatorica 12, 303–316 (1992). https://doi.org/10.1007/BF01285819

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AMS subject classification code (1991)

  • 60 C 05
  • 05 A 20