On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime

Abstract

Given a finite family $\mathcal{F}$ of linear forms with integer coefficients, and a compact abelian group G, an $\mathcal{F}$-free set in G is a measurable set which does not contain solutions to any equation L(x)=0 for L in $\mathcal{F}$. We denote by $d_{\mathcal{F}}(G)$ the supremum of μ(A) over $\mathcal{F}$-free sets A⊂G, where μ is the normalized Haar measure on G. Our main result is that, for any such collection $\mathcal{F}$ of forms in at least three variables, the sequence $d_{\mathcal{F}}({\mathbb {Z}}_{p})$ converges to $d_{\mathcal{F}}({\mathbb {R}}/{\mathbb {Z}})$ as p→∞ over primes. This answers an analogue for ℤ _p of a question that Ruzsa raised about sets of integers.