New examples of noncommutative $Łambda(p)$ sets

Abstract

In this paper, we introduce a certain combinatorial property $Z^\star(k)$, which is defined for every integer $k\ge 2$, and show that every set $E\subset\Z$ with the property $Z^\star(k)$ is necessarily a noncommutative $\Lambda(2k)$ set. In particular, using number theoretic results about the number of solutions to so-called ``$S$-unit equations,'' we show that for any finite set $Q$ of prime numbers the set $E_Q$ of natural numbers whose prime divisors all lie in the set $Q$ is noncommutative $\Lambda(p)$ for every real number $2<p<\infty$.