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
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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 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
  • Additive combinatorics

AMS Classification

  • 11B75

References

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Cite this paper

APA

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

Chicago

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

MLA

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

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