Illinois Journal of Mathematics

New examples of noncommutative $Łambda(p)$ sets

William D. Banks and Asma Harcharras

Full-text: Open access


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

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

First available in Project Euclid: 13 November 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L52: Noncommutative function spaces
Secondary: 11N25: Distribution of integers with specified multiplicative constraints 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) 47B10: Operators belonging to operator ideals (nuclear, p-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20]


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.

Export citation