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Sums and Products of Distinct Sets and Distinct Elements in ℂ

Published/Copyright: November 5, 2010
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Integers
From the journal Volume 10 Issue 5

Abstract

Let A and B be finite subsets of ℂ such that |B| = C|A|. We show the following variant of the sum product phenomenon: If |AB| < α|A| and α ≪ log |A|, then |k A + l B| ≫ |A|k|B|l. This is an application of a result of Evertse, Schlickewei, and Schmidt on linear equations with variables taking values in multiplicative groups of finite rank, in combination with an earlier theorem of Ruzsa about sumsets in . As an application of the case A = B we give a lower bound on |A+|+|A×|, where A+ is the set of sums of distinct elements of A and A× is the set of products of distinct elements of A.

Keywords.: Sum-product problem; sumset; Freiman's theorem; subspace theorem; multiplicative dimension
Received: 2009-03-11
Revised: 2010-01-05
Accepted: 2010-07-01
Published Online: 2010-11-05
Published in Print: 2010-November

© de Gruyter 2010

Articles in the same Issue

  1. Some Divisibility Properties of Binomial Coefficients and the Converse of Wolstenholme's Theorem
  2. On the Iteration of a Function Related to Euler's φ-Function
  3. An Explicit Evaluation of the Gosper Sum
  4. On the Frobenius Problem for {ak, ak + 1, ak + a, . . . , ak + ak−1}
  5. Generalizing the Combinatorics of Binomial Coefficients via -Nomials
  6. Finding Almost Squares V
  7. On Relatively Prime Subsets and Supersets
  8. Reformed Permutations in Mousetrap and Its Generalizations
  9. On Ternary Inclusion-Exclusion Polynomials
  10. Sums and Products of Distinct Sets and Distinct Elements in ℂ
https://doi.org/10.1515/integ.2010.049
Keywords for this article
Sum-product problem; sumset; Freiman's theorem; subspace theorem; multiplicative dimension

Articles in the same Issue

  1. Some Divisibility Properties of Binomial Coefficients and the Converse of Wolstenholme's Theorem
  2. On the Iteration of a Function Related to Euler's φ-Function
  3. An Explicit Evaluation of the Gosper Sum
  4. On the Frobenius Problem for {ak, ak + 1, ak + a, . . . , ak + ak−1}
  5. Generalizing the Combinatorics of Binomial Coefficients via -Nomials
  6. Finding Almost Squares V
  7. On Relatively Prime Subsets and Supersets
  8. Reformed Permutations in Mousetrap and Its Generalizations
  9. On Ternary Inclusion-Exclusion Polynomials
  10. Sums and Products of Distinct Sets and Distinct Elements in ℂ
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