2,987 research outputs found
Additive Decompositions of Subgroups of Finite Fields
We say that a set is additively decomposed into two sets and , if
. Here we study additively decompositions of
multiplicative subgroups of finite fields. In particular, we give some
improvements and generalisations of results of C. Dartyge and A. Sarkozy on
additive decompositions of quadratic residues and primitive roots modulo .
We use some new tools such the Karatsuba bound of double character sums and
some results from additive combinatorics
Counting Additive Decompositions of Quadratic Residues in Finite Fields
We say that a set is additively decomposed into two sets and if
. A. S\'ark\"ozy has recently conjectured that
the set of quadratic residues modulo a prime does not have nontrivial
decompositions. Although various partial results towards this conjecture have
been obtained, it is still open. Here we obtain a nontrivial upper bound on the
number of such decompositions
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
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