Large sets in finite fields are sumsets

✍ Scribed by Noga Alon

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 132 KB
• Volume: 126
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2006.11.007

✦ Synopsis

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 pf (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) 1 9 log 2 p and proved, with Gowers, that f (p) < cp 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,

log 1/3 p .

The proofs combine probabilistic arguments with spectral techniques.