By Enrico Bombieri (auth.), Michael Benedicks, Peter W. Jones, Stanislav Smirnov, Björn Winckler (eds.)

ISBN-10: 3540304320

ISBN-13: 9783540304326

ISBN-10: 3540304347

ISBN-13: 9783540304340

The goal of the essays amassed during this quantity is to think about the way forward for research and comparable parts of physics. it truly is released in honor of Lennart Carleson, who has dedicated a lot of his profession to broadening the scope of harmonic research. Written via best mathematicians and mathematical physicists, the articles should still motivate new avenues of study. the gathering indicates the notable dating among actual instinct and mathematical research. those essays provide self belief that the assembly floor among those parts of arithmetic and physics will in simple terms develop extra fertile within the future.

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Additional resources for Perspectives in Analysis: Essays in Honor of Lennart Carleson’s 75th Birthday

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1 Uniform Distribution of Mersenne Numbers The Mersenne numbers are defined as Mq = 2q − 1, where q is prime. 1. ,t2+ε q prime is uniformly distributed, t = multiplicative order of 2 modulo p. 5. This result was established earlier in [9], assuming GRH. 2 Prime Divisors of ‘Sparse Integers’ For g ≥ 2 and s ≥ 1 two integers and D = {di }si=0 a sequence of s+ 1 non-zero integers, denote Sg,s (D) = {d0 + d1 g m1 + · · · + ds g ms | mi ∈ Z+ } . 2 ([5]). For all δ > 0, there is a s(δ) such that if s > s(δ), for most primes p, there is a n ∈ Sg,s (D) such that p | n and log n < pδ .

The Szemeredi–Trotter theorem implies that |S|2 > cN 2 N 3 ⇒ |S| > cN 5/2 (using only that L consists of distinct lines). Small improvement known: |S| > N 5/2+δ . 2 Algebraic Setting Let S ⊂ F3p (p prime) contain a line in every direction. Is it true that |S| p3−ε ? It is known (see [12]) that |S| > p5/2+δ . 4 The Relation with Sum-product Problems This development originates from [23]. It appears that the Kakeya (and related) conjectures require a better understanding of purely one-dimensional issues, basically expressing that either the sum or the product set of a given set A is significantly larger than A itself.

T. Tao: Geom. Funct. Anal. 13 no 6, 1359 (2003) 28. Th. Varopoulos, L. Saloff-Coste, T. Coulhon: Analysis and geometry on groups (Cambridge University Press, Cambridge 1992) 29. T. Wolff: Rev. Mat. Iberoamericana 11 no 3, 651 (1995) 30. T. Wolff: Maximal averages and packing of one-dimensional sets. In: Proceedings of International Congress of Mathematicians, vol II (Berlin 1998) pp 755–761 31. T. Wolff: Recent work connected with the Kakeya problem. R. edu 1 Introduction Our goal is to describe some of the mathematical challenges confronted, when dealing with massive data sets.

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Perspectives in Analysis: Essays in Honor of Lennart Carleson’s 75th Birthday by Enrico Bombieri (auth.), Michael Benedicks, Peter W. Jones, Stanislav Smirnov, Björn Winckler (eds.)

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