Download e-book for kindle: Additive combinatorics by Terence Tao

By Terence Tao

ISBN-10: 0521853869

ISBN-13: 9780521853866

Additive combinatorics is the idea of counting additive buildings in units. This idea has visible fascinating advancements and dramatic adjustments in path lately because of its connections with parts similar to quantity concept, ergodic thought and graph thought. This graduate point textual content will enable scholars and researchers effortless access into this interesting box. the following, for the 1st time, the authors collect in a self-contained and systematic demeanour the various various instruments and ideas which are utilized in the trendy concept, providing them in an available, coherent, and intuitively transparent demeanour, and supplying fast purposes to difficulties in additive combinatorics. the ability of those instruments is easily validated within the presentation of contemporary advances similar to Szemerédi's theorem on mathematics progressions, the Kakeya conjecture and Erdos distance difficulties, and the constructing box of sum-product estimates. The textual content is supplemented through numerous workouts and new effects.

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Tn ), and denote the order of α as |α| := α1 + · · · + αn . For any order d ≥ 0, we denote Ed (Y ) := maxα:|α|=d E(∂ α Y ); thus for instance E0 (Y ) = E(Y ), and Ed (Y ) = 0 if d exceeds the degree of Y . These quantities are vaguely reminiscent of Sobolev norms for the random variable Y . We also define E≥d (Y ) := maxd ≥d Ed (Y ). The following result is due to Kim and Vu [203]. 36 Let k ≥ 1, and let Y = Y (t1 , . . , tn ) be a totally positive polynomial of n independent boolean variables t1 , .

Tn ) of degree k; more explicitly, we have Y (t1 , . . , tn ) = tj A∈An j∈A where An is the collection of all sets {x1 , . . 1 < x1 < x2 < · · · < xk . 43 we see that E(Y ) = k (C log n) when n is sufficiently large depending on C, k. To conclude the proof it would thus suffice by the Borel–Cantelli lemma to establish the large deviation inequality P |Y − E(Y )| > 1 E(Y ) = OC,k 2 1 n2 for all large n. 37 (and choosing C sufficiently large), we see that it suffices to show the derivative estimates E1 (Y ), .

9 Thin Waring bases Recall that a thin basis of order k is a set B ⊂ N such that rk,B (n) = O(log n) for all large n. 15, proved above, asserts that N contains a thin basis of any order. 46 Let A be any fixed basis of order k. Does A contain a thin subbasis B? Note that Sidon’s original question can be viewed as the k = 2, A = N case of this question. 21) we know that a thin basis B enjoys the bounds |B ∩ [0, N ]| = k N 1/k ; |B ∩ [0, N ]| = Ok N 1/k log1/k N for all large N . 47 Let A be any fixed basis of order k.

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Additive combinatorics by Terence Tao


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