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

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## Details

### Authors and affiliations

Raj Kumar Mistri

*Department of Mathematics, Harish-Chandra Research Institute, HBNI
Chhatnag Road, Jhunsi, Allahabad – 211 019, India
*

### Abstract

Let *A*⊆ *Z* and *B* ⊆ *Z* 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* : *a* ∈ *A* and *b* ∈ *B*}. 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* + *r · B *for two sets

*A*and

*B*, where

*r*is an integer with |

*r*| ≥ 3.

### Keywords

- Sum of dilates
- Minkowski sumsets
- Sum-product problem
- Additive combinatorics

### AMS Classification

- 11B75

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## Related papers

- Mistri, R. K. (2018) Corrigendum to “Sum of dilates of two sets” [Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 4, 34–41]. Notes on Number Theory and Discrete Mathematics, 24(1), 136.

## Cite this paper

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

ChicagoMistri, Raj Kumar. “Sum of Dilates of Two Sets.” Notes on Number Theory and Discrete Mathematics 23, no. 4 (2017): 34-41.

MLAMistri, Raj Kumar. “Sum of Dilates of Two Sets.” Notes on Number Theory and Discrete Mathematics 23.4 (2017): 34-41. Print.