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A Lower Bound for the Size of a Minkowski Sum of Dilates

  • Y. O. HAMIDOUNE (a1) and J. RUÉ (a2)

Let A be a finite non-empty set of integers. An asymptotic estimate of the size of the sum of several dilates was obtained by Bukh. The unique known exact bound concerns the sum |A + kA|, where k is a prime and |A| is large. In its full generality, this bound is due to Cilleruelo, Serra and the first author.

Let k be an odd prime and assume that |A| > 8kk. A corollary to our main result states that |2⋅A + kA|≥(k+2)|A|−k2k+2. Notice that |2⋅P+kP|=(k+2)|P|−2k, if P is an arithmetic progression.

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[1]Bukh, B. (2008) Non-trivial solutions to a linear equation in integers. Acta Arithmetica 131 4155.
[2]Bukh, B. (2008) Sums of dilates. Combin. Probab. Comput. 17 627639.
[3]Cilleruelo, J., Hamidoune, Y. O. and Serra, O. (2009) On sums of dilates. Combin. Probab. Comput. 18 871880.
[4]Cilleruelo, J., Silva, M. and Vinuesa, C. (2010) A sumset problem. J. Combin. Number Theory 2.
[5]Garaev, M. Z. (2007) An explicit sum-product estimate in p. Internat. Math. Res. Notices 2007 #rnm035.
[6]Katz, N. H. and Shen, C.-Y. (2008) A slight improvement to Garaev's sum product estimate. Proc. Amer. Math. Soc. 136 24992504.
[7]Łaba, I. and Konyagin, S. (2006) Distance sets of well-distributed planar sets for polygonal norms. Israel J. Math. 152 157179.
[8]Nathanson, M. B. Inverse problems for linear forms over finite sets of integers. Available online at: arXiv: 0708.2304v2.
[9]Nathanson, M. B., O'Bryant, K., Orosz, B., Ruzsa, I. and Silva, M. (2007) Binary linear forms over finite sets of integers. Acta Arithmetica 129 341361.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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