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From harmonic analysis to arithmetic combinatorics. (English) Zbl 1160.11004

The paper under review is an expository overview of the exciting recent links between harmonic analysis and arithmetic combinatorics. The author’s perspective comes from harmonic analysis which is most welcome and the paper begins with the Kakeya conjecture and the Restriction phenomenon before moving on to combinatorial geometry and then quite a bit of additive combinatorics (including Freĭman’s theorem, the sum-product phenomenon and Szemerédi’s theorem). The paper culminates in a discussion of the celebrated theorem of Green and Tao that the primes contain arbitrarily long arithmetic progressions.
The paper includes many interesting open problems and the author’s bent should make it particularly enlightening to people in arithmetic combinatorics looking to diversify. The writing style is entertaining and the mathematics is peppered with interesting historical asides helping to animate the material with a human dimension. There is an extensive bibliography.

MSC:

11B25 Arithmetic progressions
11B75 Other combinatorial number theory
11L07 Estimates on exponential sums
28A75 Length, area, volume, other geometric measure theory
28A78 Hausdorff and packing measures
42B15 Multipliers for harmonic analysis in several variables
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
52C10 Erdős problems and related topics of discrete geometry
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References:

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