On a Kakeya-type problem

Gregory A. Freiman, Yonutz V. Stanchescu

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Let A be a flnite subset of an abelian group G. For every element bi of the sumset 2A=b0, b1,.., b2A-1 we denote by Di=a-a′. a, a′ϵA, a+a′=bi and ri=(a, a′). a+a′=bi, a, a′ϵA. After an eventual reordering of 2A, we may assume that r0≥r1≥..≥r2A-1. For every 1≤s≤2A we deflne Rs(A) = D0[D1[..[Ds-1 and Rs(k)= maxRs(A). AG, A=kg. Bourgain and Katz and Tao obtained an estimate of Rs(k) assuming s being of order k. In this note we flnd the exact value of Rs(k) in cases s=1, s=2 and s=3. The case s=3 appeared to be not simple. The structure of extremal sets led us to sets isomorphic to planar sets having a rather unexpected form of a perfect hexagon. The proof suggests the way of dealing with the general case s≥4.

Original languageEnglish
Pages (from-to)131-148
Number of pages18
JournalFunctiones et Approximatio, Commentarii Mathematici
Volume37
Issue number1
DOIs
StatePublished - 2007

Keywords

  • Inverse additive number theory
  • Kakeya problem

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