Sumsets Containing Powers of an Integer

Let n≥1, and let m be an integer with m≥2. We show that if a subset A of the interval [0,n] satisfies that 0∈A and |A|>1+n/2, then mA, the set of the sum of m (not necessarily distinct) elements in A, has a power of m. This result is best possible in the case that m is odd.

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Correspondence to Tetsuya Abe.

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Abe, T. Sumsets Containing Powers of an Integer. Combinatorica 24, 1–4 (2004).

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Mathematics Subject Classification (2000):

  • 11B13
  • 05D05