208 research outputs found
Restricted sumsets in multiplicative subgroups
We establish the restricted sumset analog of the celebrated conjecture of
S\'{a}rk\"{o}zy on additive decompositions of the set of nonzero squares over a
finite field. More precisely, we show that if is an odd prime power,
then the set of nonzero squares in cannot be written as a
restricted sumset , extending a result of Shkredov. More
generally, we study restricted sumsets in multiplicative subgroups over finite
fields as well as restricted sumsets in perfect powers (over integers)
motivated by a question of Erd\H{o}s and Moser. We also prove an analog of van
Lint-MacWilliams' conjecture for restricted sumsets, equivalently, an analog of
Erd\H{o}s-Ko-Rado theorem in a family of Cayley sum graphs.Comment: 23 page
On various restricted sumsets
For finite subsets A_1,...,A_n of a field, their sumset is given by
{a_1+...+a_n: a_1 in A_1,...,a_n in A_n}. In this paper we study various
restricted sumsets of A_1,...,A_n with restrictions of the following forms:
a_i-a_j not in S_{ij}, or alpha_ia_i not=alpha_ja_j, or a_i+b_i not=a_j+b_j
(mod m_{ij}). Furthermore, we gain an insight into relations among recent
results on this area obtained in quite different ways.Comment: 11 pages; final version for J. Number Theor
Three-term arithmetic progressions and sumsets
Suppose that G is an abelian group and A is a finite subset of G containing
no three-term arithmetic progressions. We show that |A+A| >> |A|(log
|A|)^{1/3-\epsilon} for all \epsilon>0.Comment: 20 pp. Corrected typos. Updated references. Corrected proof of
Theorem 5.1. Minor revisions
Almost all primes have a multiple of small Hamming weight
Recent results of Bourgain and Shparlinski imply that for almost all primes
there is a multiple that can be written in binary as with or ,
respectively. We show that (corresponding to Hamming weight )
suffices.
We also prove there are infinitely many primes with a multiplicative
subgroup , for some
, of size , where the sum-product set
does not cover completely
Additive structures in sumsets
Suppose that A is a subset of the integers {1,...,N} of density a. We provide
a new proof of a result of Green which shows that A+A contains an arithmetic
progression of length exp(ca(log N)^{1/2}) for some absolute c>0. Furthermore
we improve the length of progression guaranteed in higher sumsets; for example
we show that A+A+A contains a progression of length roughly N^{ca} improving on
the previous best of N^{ca^{2+\epsilon}}.Comment: 28 pp. Corrected typos. Updated references
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