New examples of noncommutative Λ(p) sets

Abstract

In this paper, we introduce a certain combinatorial property Z*(k), which is defined for every integer k ≥ 2, and show that every set E ⊂ Z with the property Z*(k) is necessarily a noncommutative Λ (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, EQ is noncommutative Λ(p) for every real number 2 < p < ∞, where EQ is the set of natural numbers whose prime divisors all lie in the set Q.