The Central Limit Theorem for Subsequences in Probabilistic Number Theory

Abstract

Abstract Let ( nk ) k ≥1 be an increasing sequence of positive integers, and let f ( x ) be a real function satisfying If the distribution of converges to a Gaussian distribution. In the case there is a complex interplay between the analytic properties of the function f , the number-theoretic properties of ( nk ) k ≥1 , and the limit distribution of (2). In this paper we prove that any sequence ( nk ) k ≥1 satisfying lim contains a nontrivial subsequence ( mk ) k ≥1 such that for any function satisfying (1) the distribution of converges to a Gaussian distribution. This result is best possible: for any ε > 0 there exists a sequence ( nk ) k ≥1 satisfying lim such that for every nontrivial subsequence ( mk ) k ≥1 of ( nk ) k ≥1 the distribution of (2) does not converge to a Gaussian distribution for some f . Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

Locations

• Canadian Journal of Mathematics - View - PDF