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Sums of Dilates in p

Published online by Cambridge University Press:  30 October 2012

GONZALO FIZ PONTIVEROS
GONZALO FIZ PONTIVEROS*
Affiliation:
IMPA, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, Brasil (e-mail: [email protected])
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Abstract

We consider the problem of sums of dilates in groups of prime order. It is well known that sets with small density and small sumset in p behave like integer sets. Thus, given A ⊂ p of sufficiently small density, it is straightforward to show that

\begin{linenomath} $$| \lambda_{1}A+\lambda_{2}A+\cdots+ \lambda_{k}A | \ge\biggl(\sum_{i}|\lambda_{i}|\biggr)|A|- o(|A|).$$ \end{linenomath}
On the other hand, the behaviour for sets of large density turns out to be rather surprising. Indeed, for any ε > 0, we construct subsets of density 1/2–ε such that |A + λ A| ≤ (1–δ)p, showing that there is a very different behaviour for subsets of large density.

Keywords

Primary 11B1305B10
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 2 , March 2013 , pp. 282 - 293
Copyright
Copyright © Cambridge University Press 2012

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