5,407 research outputs found
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
Integrability of oscillatory functions on local fields: transfer principles
For oscillatory functions on local fields coming from motivic exponential
functions, we show that integrability over implies integrability over
for large , and vice versa. More generally, the integrability
only depends on the isomorphism class of the residue field of the local field,
once the characteristic of the residue field is large enough. This principle
yields general local integrability results for Harish-Chandra characters in
positive characteristic as we show in other work. Transfer principles for
related conditions such as boundedness and local integrability are also
obtained. The proofs rely on a thorough study of loci of integrability, to
which we give a geometric meaning by relating them to zero loci of functions of
a specific kind.Comment: 44 page
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