## Illinois Journal of Mathematics

### 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 $2<p<\infty$.

#### Article information

Source
Illinois J. Math., Volume 47, Number 4 (2003), 1063-1078.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138091

Digital Object Identifier
doi:10.1215/ijm/1258138091

Mathematical Reviews number (MathSciNet)
MR2036990

Zentralblatt MATH identifier
1035.43004

#### Citation

Banks, William D.; Harcharras, Asma. New examples of noncommutative $Łambda(p)$ sets. Illinois J. Math. 47 (2003), no. 4, 1063--1078. doi:10.1215/ijm/1258138091. https://projecteuclid.org/euclid.ijm/1258138091