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14
EXTENSIONS OF THE STEINTOMAS THEOREM
, 2010
"... We prove an endpoint version of the SteinTomas restriction theorem, for a general class of measures, and with a strengthened Lorentz space estimate. A similar improvement is obtained for Stein’s estimate on oscillatory integrals of CarlesonSjölinHörmander type and some spectral projection opera ..."
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We prove an endpoint version of the SteinTomas restriction theorem, for a general class of measures, and with a strengthened Lorentz space estimate. A similar improvement is obtained for Stein’s estimate on oscillatory integrals of CarlesonSjölinHörmander type and some spectral projection operators on compact manifolds, and for classes of oscillatory integral operators with onesided fold singularities.
Finite configurations in sparse sets
, 2013
"... Let E ⊆ Rn be a closed set of Hausdorff dimension α. For m ≥ n, let {B1,..., Bk} be n × (m − n) matrices. We prove that if the system of matrices Bj is nondegenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fou ..."
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Let E ⊆ Rn be a closed set of Hausdorff dimension α. For m ≥ n, let {B1,..., Bk} be n × (m − n) matrices. We prove that if the system of matrices Bj is nondegenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of m depending on n and k, the set E contains a translate of a nontrivial kpoint configuration {B1y,..., Bky}. As a consequence, we are able to establish existence of certain geometric configurations in Salem sets (such as parallelograms in R n and isosceles right triangles in R 2). This can be viewed as a multidimensional analogue of
Sets of salem type and sharpness of the l2fourier restriction theorem
 Transactions of the American Mathematical Society
, 2013
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Additive Combinatorics with a view towards Computer Science and Cryptography  An Exposition
, 2011
"... Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define – perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is ..."
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Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define – perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is a branch of mathematics concerning the study of additive structures in sets equipped with a group structure – we may have other structure that interacts with this group structure. This newly emerging field has seen tremendous advances over the last few years, and has recently become a focus of attention among both mathematicians and computer scientists. This fascinating area has been enriched by its formidable links to combinatorics, number theory, harmonic analysis, ergodic theory, and some other branches; all deeply crossfertilize each other, holding great promise for all of them! There is a considerable number of incredible problems, results, and novel applications in this thriving area. In this exposition, we attempt to provide an illuminating overview of some conspicuous breakthroughs in this captivating field, together with a number of seminal applications to sundry parts of mathematics and some other disciplines, with emphasis on computer science and cryptography.
CONVOLUTION POWERS OF SALEM MEASURES WITH APPLICATIONS
"... Abstract. We study the regularity of convolution powers for measures supported on Salem sets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for α of the form d/n, n = 2, 3,... there exist αSalem measures for which the L2 Fourier restriction th ..."
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Abstract. We study the regularity of convolution powers for measures supported on Salem sets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for α of the form d/n, n = 2, 3,... there exist αSalem measures for which the L2 Fourier restriction theorem holds in the range p ≤ 2d 2d−α. The results rely on ideas of Körner. We extend some of his constructions to obtain upper regular αSalem measures, with sharp regularity results for nfold convolutions for all n ∈ N. 1.