270 research outputs found
Iterated Sumsets and Subsequence Sums
Let
be a finite abelian group with . The Kemperman
Structure Theorem characterizes all subsets satisfying
and has been extended to cover the case when . Utilizing these results, we provide a precise structural description
of all finite subsets with when
(also when is infinite), in which case many of the pathological
possibilities from the case vanish, particularly for large . The structural description is combined with other arguments to
generalize a subsequence sum result of Olson asserting that a sequence of
terms from having length must either have every element of
representable as a sum of -terms from or else have all but
of its terms lying in a common -coset for some . We show
that the much weaker hypothesis suffices to obtain a
nearly identical conclusion, where for the case is trivial we must allow
all but terms of to be from the same -coset. The bound on
is improved for several classes of groups , yielding optimal lower
bounds for . We also generalize Olson's result for -term subsums to
an analogous one for -term subsums when , with the bound
likewise improved for several special classes of groups. This improves previous
generalizations of Olson's result, with the bounds for optimal.Comment: Revised version, with results reworded to appear less technica
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
Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group
We inspect the relationship between relative Fourier multipliers on
noncommutative Lebesgue-Orlicz spaces of a discrete group and relative
Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four
applications are given: lacunary sets; unconditional Schauder bases for the
subspace of a Lebesgue space determined by a given spectrum, that is, by a
subset of the group; the norm of the Hilbert transform and the Riesz projection
on Schatten-von-Neumann classes with exponent a power of 2; the norm of
Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less
than 1.Comment: Corresponds to the version published in the Canadian Journal of
Mathematics 63(5):1161-1187 (2011
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