Abstract
Let S be a set of integers or of residue classes modulo a prime p, with cardinality |S| = k, and let T be the set of all sums of two distinct elements of S. For the integer case, it is shown that if |T| is less than approximately 2.5k then S is contained in an arithmetic progression with relatively small cardinality. For the residue class case a result of this type is derived provided that k > 60 and p > 50k. As an application, it is shown that |T| ≥ 2k-3 under these conditions. Earlier results of Freiman play an essential role in the proofs.
Original language | English |
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Pages (from-to) | 163-172 |
Number of pages | 10 |
Journal | Asterisque |
Volume | 258 |
State | Published - 1999 |
Keywords
- Set addition
- Sums of residues
- Sumset