151 research outputs found
A sum-product estimate in fields of prime order
Let q be a prime, A be a subset of a finite field ,
. We prove the estimate for some and c>0. This extends the result of J.
Bourgain, N. Katz, and T. Tao
Two -unit equations with many solutions
We show that there exist arbitrarily large sets of prime numbers such
that the equation has more than
solutions in coprime integers , , all of whose prime factors lie in
the set . We also show that there exist sets for which the equation
has more than solutions with all prime factors
of and lying in .Comment: 6 page
New bounds for Gauss sums derived from -th powers, and for Heilbronn's exponential sum
We show that ∑pn=1exp(2πiank/p) ≪ min(k5/8p5/8, k3/8p3/4) and ∑pn=1exp(2πianp/p2) ≪ p7/8 when p \(crossed)a. The proof uses a modification of Stepanov's method
Delta-semidefinite and delta-convex quadratic forms in Banach spaces
A continuous quadratic form ("quadratic form", in short) on a Banach space
is: (a) delta-semidefinite (i.e., representable as a difference of two
nonnegative quadratic forms) if and only if the corresponding symmetric linear
operator factors through a Hilbert space; (b) delta-convex
(i.e., representable as a difference of two continuous convex functions) if and
only if is a UMD-operator. It follows, for instance, that each quadratic
form on an infinite-dimensional space () is: (a)
delta-semidefinite iff ; (b) delta-convex iff . Some other
related results concerning delta-convexity are proved and some open problems
are stated.Comment: 19 page
Greedy Approximation with Regard to Bases and General Minimal Systems
*This research was supported by the National Science Foundation Grant DMS 0200187 and by ONR Grant N00014-96-1-1003This paper is a survey which also contains some new results on
the nonlinear approximation with regard to a basis or, more generally, with
regard to a minimal system. Approximation takes place in a Banach or in
a quasi-Banach space. The last decade was very successful in studying nonlinear
approximation. This was motivated by numerous applications. Nonlinear
approximation is important in applications because of its increased
efficiency. Two types of nonlinear approximation are employed frequently
in applications. Adaptive methods are used in PDE solvers. The m-term
approximation considered here is used in image and signal processing as well
as the design of neural networks. The basic idea behind nonlinear approximation
is that the elements used in the approximation do not come from
a fixed linear space but are allowed to depend on the function being approximated.
The fundamental question of nonlinear approximation is how
to construct good methods (algorithms) of nonlinear approximation. In this
paper we discuss greedy type and thresholding type algorithms
Explicit constructions of RIP matrices and related problems
We give a new explicit construction of matrices satisfying the
Restricted Isometry Property (RIP). Namely, for some c>0, large N and any n
satisfying N^{1-c} < n < N, we construct RIP matrices of order k^{1/2+c}. This
overcomes the natural barrier k=O(n^{1/2}) for proofs based on small coherence,
which are used in all previous explicit constructions of RIP matrices. Key
ingredients in our proof are new estimates for sumsets in product sets and for
exponential sums with the products of sets possessing special additive
structure. We also give a construction of sets of n complex numbers whose k-th
moments are uniformly small for 1\le k\le N (Turan's power sum problem), which
improves upon known explicit constructions when (\log N)^{1+o(1)} \le n\le
(\log N)^{4+o(1)}. This latter construction produces elementary explicit
examples of n by N matrices that satisfy RIP and whose columns constitute a new
spherical code; for those problems the parameters closely match those of
existing constructions in the range (\log N)^{1+o(1)} \le n\le (\log
N)^{5/2+o(1)}.Comment: v3. Minor correction
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