## Abstract

For a prime p, a subset S of Z_{p} is a sumset if S = A + A for some A ⊂ Z_{p}. Let f (p) denote the maximum integer so that every subset S ⊂ Z_{p} 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) log_{2} p and proved, with Gowers, that f (p) < c p^{2 / 3} log^{1 / 3} p for some absolute constant c. Here we improve these estimates, showing that there are two absolute positive constants c_{1}, c_{2} so that for all sufficiently large p,c_{1} frac(sqrt(p), sqrt(log p)) ≤ f (p) < c_{2} frac(p^{2 / 3}, log^{1 / 3} p) . The proofs combine probabilistic arguments with spectral techniques.

Original language | English (US) |
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Pages (from-to) | 110-118 |

Number of pages | 9 |

Journal | Journal of Number Theory |

Volume | 126 |

Issue number | 1 |

DOIs | |

State | Published - Sep 2007 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

## Keywords

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