Three-term arithmetic progressions and sumsets
نویسندگان
چکیده
منابع مشابه
Three Term Arithmetic Progressions in Sumsets
Suppose that G is an abelian group and A ⊂ G is finite and contains no non-trivial three term arithmetic progressions. We show that |A + A| ≫ε |A|(log |A|) 1 3 −ε.
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In this paper we show that sumsets A + B of finite sets A and B of integers, must contain long arithmetic progressions. The methods we use are completely elementary, in contrast to other works, which often rely on harmonic analysis. –Dedicated to Ron Graham on the occasion of his 70 birthday
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One of the main focuses in combinatorial (and additive) number theory is that of “understanding” the structure of the sumset A + B = {a + b : a ∈ A, b ∈ B}, given certain information about the sets A and B. For example, one such problem is to determine the length of the longest arithmetic progression in this sumset, given that A,B ⊆ {0, 1, 2, ..., N} and |A|, |B| > δN , for some 0 < δ ≤ 1. The ...
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Let A, B ⊆ Z be finite, nonempty subsets such that maxB − minB ≤ maxA − minA, gcd(A+B−c) = 1, for some c ∈ A+B, and |A+B| ≤ |A|+2|B|−3−δ(A,B), where δ(A,B) is 1 if x + A ⊆ B for some x ∈ Z, and is 0 otherwise. Assume one of the following conditions holds true: • maxA−minA ≤ |A| + |B|− 3, • gcd(A− a) ≤ 2, for some a ∈ A, • |A + B| ≤ 2|A| + |B|− 3− δ(B,A). Then A+B contains a (|A|+ |B|−1)–term ar...
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ژورنال
عنوان ژورنال: Proceedings of the Edinburgh Mathematical Society
سال: 2009
ISSN: 0013-0915,1464-3839
DOI: 10.1017/s0013091506001398