On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime
Combinatorics
2011-09-15 v1 Classical Analysis and ODEs
Abstract
Given a finite family F of linear forms with integer coefficients, and a compact abelian group G, an F-free set in G is a measurable set which does not contain solutions to any equation L(x)=0 for L in F. We denote by d_F(G) the supremum of m(A) over F-free sets A in G, where m is the normalized Haar measure on G. Our main result is that, for any such collection F of forms in at least three variables, the sequence d_F(Z_p) converges to d_F(R/Z) as p tends to infinity over primes. This answers an analogue for Z_p of a question that Ruzsa raised about sets of integers.
Cite
@article{arxiv.1109.2934,
title = {On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime},
author = {Pablo Candela and Olof Sisask},
journal= {arXiv preprint arXiv:1109.2934},
year = {2011}
}
Comments
17 pages