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SUMSETS AND DIFFERENCE SETS CONTAINING A COMMON TERM OF A SEQUENCE

  • QUAN-HUI YANG (a1) and YONG-GAO CHEN (a2)
Abstract
Abstract

Let β>1 be a real number, and let {ak} be an unbounded sequence of positive integers such that ak+1/akβ for all k≥1. The following result is proved: if n is an integer with n>(1+1/(2β))a1 and A is a subset of {0,1,…,n} with , then (A+A)∩(AA) contains a term of {ak }. The lower bound for |A| is optimal. Beyond these, we also prove that if n≥3 is an integer and A is a subset of {0,1,…,n} with , then (A+A)∩(AA) contains a power of 2. Furthermore, cannot be improved.

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Copyright
Corresponding author
For correspondence; e-mail: ygchen@njnu.edu.cn
Footnotes
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This work was supported by the National Natural Science Foundation of China, Grant No. 11071121.

Footnotes
References
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[1]Abe T., ‘Sumsets containing powers of an integer’, Combinatorica 24 (2004), 14.
[2]Alon N., ‘Subset sums’, J. Number Theory 27 (1987), 196205.
[3]Erdős P. and Freiman G., ‘On two additive problems’, J. Number Theory 34 (1990), 112.
[4]Freiman G. A., ‘Sumsets and powers of 2’, Colloq. Math. Soc. János Bolyai 60 (1992), 279286.
[5]Kapoor V., ‘Sets whose sumset avoids a thin sequence’, J. Number Theory 130 (2010), 534538.
[6]Lev V. F., ‘Representing powers of 2 by a sum of four integers’, Combinatorica 16 (1996), 413416.
[7]Nathanson M. B. and Sárközy A., ‘Sumsets containing long arithmetic progressions and powers of 2’, Acta Arith. 54 (1989), 147154.
[8]Pan H., ‘Note on integer powers in sumsets’, J. Number Theory 117 (2006), 216221.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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