Abstract
We show that for any integer n ≥ 1 and real ε > 0, the union of n arithmetic progressions with pairwise distinct differences, each of length n, contains at least c (ε)n2-ε elements, where c (ε) is a positive constant depending only on ε. This estimate is sharp in the sense that the assertion becomes invalid for ε = 0. We also obtain estimates for the "asymmetric case" where the number of progressions is distinct from their lengths.
Original language | English |
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Pages (from-to) | 1062-1073 |
Number of pages | 12 |
Journal | SIAM Journal on Discrete Mathematics |
Volume | 28 |
Issue number | 3 |
DOIs | |
State | Published - 2014 |
Bibliographical note
Publisher Copyright:© 2014 Society for Industrial and Applied Mathematics.
Keywords
- Additive combinatorics
- Arithmetic progressions
- Bondy-Simonovits theorem