Large sets in finite fields are sumsets

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Abstract

For a prime p, a subset S of Zp is a sumset if S = A + A for some A ⊂ Zp. Let f (p) denote the maximum integer so that every subset S ⊂ Zp of size at least p - f (p) is a sumset. The question of determining or estimating f (p) was raised by Green. He showed that for all sufficiently large p, f (p) ≥ frac(1, 9) log2 p and proved, with Gowers, that f (p) < c p2 / 3 log1 / 3 p for some absolute constant c. Here we improve these estimates, showing that there are two absolute positive constants c1, c2 so that for all sufficiently large p,c1 frac(sqrt(p), sqrt(log p)) ≤ f (p) < c2 frac(p2 / 3, log1 / 3 p) . The proofs combine probabilistic arguments with spectral techniques.

Original languageEnglish (US)
Pages (from-to)110-118
Number of pages9
JournalJournal of Number Theory
Volume126
Issue number1
DOIs
StatePublished - Sep 2007
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Keywords

  • Cayley sum graph
  • Character sums
  • Graph eigenvalues
  • Probabilistic method
  • Sumset

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