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SUMSETS AND DIFFERENCE SETS CONTAINING A COMMON TERM OF A SEQUENCE
Published online by Cambridge University Press: 26 September 2011
Abstract
Let β>1 be a real number, and let {ak} be an unbounded sequence of positive integers such that ak+1/ak≤β 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)∩(A−A) 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)∩(A−A) contains a power of 2. Furthermore, cannot be improved.
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MSC classification
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- Research Article
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- Copyright © Australian Mathematical Publishing Association Inc. 2011
Footnotes
This work was supported by the National Natural Science Foundation of China, Grant No. 11071121.
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