291 research outputs found
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
On the cardinality of sumsets in torsion-free groups
Let be finite subsets of a torsion-free group . We prove that for
every positive integer there is a such that if then
the inequality holds unless a left translate of is
contained in a cyclic subgroup. We obtain for arbitrary
torsion-free groups, and for groups with the unique product
property, where is an absolute constant. We give examples to show that
is at least quadratic in
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