This is a preprint of an article published in the Illinois Journal of Mathematics, vol.47 (2003), issue 4, pp.1063-1078.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
