A step beyond Freiman's theorem for set addition modulo a prime

2018/05/31 by Pablo Candela, Candela, Pablo, Oriol Serra +3
Mathematics · #05B10 #11B13 #11P70 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05B10 #msc:11B13 #msc:11P70

paper · pdf · doi:10.48550/arxiv.1805.12374

13 pages, 1 figure

arxiv created 2018/05/31 · arxiv updated 2018/06/01

Abstract

Freiman's 2.4-Theorem states that any set A ⊂ ℤ_p satisfying |2A| ≤ 2.4|A| - 3 and |A| < p/35 can be covered by an arithmetic progression of length at most |2A| - |A| + 1. A more general result of Green and Ruzsa implies that this covering property holds for any set satisfying |2A| ≤ 3|A| - 4 as long as the rather strong density requirement |A| < p/10²¹⁵ is satisfied. We present a version of this statement that allows for sets satisfying |2A| ≤ 2.48|A| - 7 with the more modest density requirement of |A| < p/10¹⁰.

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