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Acta Mathematica Hungarica

, Volume 132, Issue 3, pp 223–243 | Cite as

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

  • Pablo Candela
  • Olof Sisask
Article

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 AG, 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.

Keywords

sets free from solutions to linear equations Fourier analysis removal lemma 

2000 Mathematics Subject Classification

11B30 43A25 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2011

Authors and Affiliations

  1. 1.Centre for Mathematical SciencesCambridgeUK
  2. 2.School of Mathematical SciencesQueen Mary, University of LondonLondonUK

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