# Sum of dilates of two sets

Raj Kumar Mistri
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 4, Pages 34–41
Full paper (PDF, 157 Kb)
Corrigendum

## Details

### Authors and affiliations

Raj Kumar Mistri
Department of Mathematics, Harish-Chandra Research Institute, HBNI

### Abstract

Let A⊆ Z and BZ be nonempty finite sets and let r be a nonzero integer. The sumof dilates of A and B is defined as A + r · B := {a + rb : aA and bB}. Finding nontrivial lower bound for the sum of dilates is an important problem in additive combinatorics and it has applications in sum-product problems. In case of A = B, a recent result of Freiman et al. states that if r ≥ 3, then |A + r · A| ≥ 4|A| – 4. We generalize this result for the sum of dilates A + · B for two sets A and B, where r is an integer with |r| ≥ 3.

### Keywords

• Sum of dilates
• Minkowski sumsets
• Sum-product problem

• 11B75

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## Cite this paper

Mistri, R. K. (2017). Sum of dilates of two sets. Notes on Number Theory and Discrete Mathematics, 23(4), 34-41.