By Terence Tao

ISBN-10: 0521853869

ISBN-13: 9780521853866

Additive combinatorics is the speculation of counting additive buildings in units. This concept has noticeable interesting advancements and dramatic adjustments in path in recent times due to its connections with parts reminiscent of quantity conception, ergodic concept and graph conception. This graduate point textual content will permit scholars and researchers effortless access into this attention-grabbing box. the following, for the 1st time, the authors collect in a self-contained and systematic demeanour the numerous varied instruments and concepts which are utilized in the fashionable concept, proposing them in an obtainable, coherent, and intuitively transparent demeanour, and supplying fast purposes to difficulties in additive combinatorics. the facility of those instruments is definitely tested within the presentation of modern advances akin 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 by way of a lot of routines and new effects.

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**Sample text**

21) we see that if A is a Bh [g] set, then |A ∩ [0, n]| = Oh,g (n 1/ h ) for all n. 39 For any h ≥ 1 and > 0, there exists a set A ⊂ Z+ with |A ∩ [0, n]| = h (n 1/ h− ) for all large n, which is a Bh [g] set for some g = gh, (or in other words, rh,A (n) is uniformly bounded in n). 38 we can give a short proof of this theorem. As before, we construct A randomly, letting the events n ∈ A be independent with probability P(n ∈ A) = n 1/ h−1− . 9 and the Borel–Cantelli lemma also gives |A ∩ [0, n]| = h, (n 1/ h− ) for all but finitely many n with probability 1.

Notice that the assumption implies that Y has small expectation. Taking α to be all zero, we have E(Y ) ≤ n −γ . 36 relies on the so-called “divide and conquer martingale” technique, together with the exponential moment method. It is not too technical but requires lots of introduction. We thus skip it and refer the reader to [203]. 37 is more complicated. Besides the abovementioned martingale technique, it also requires some non-trivial combinatorial considerations. 38 is a by-product of this proof (for details see [378]).

Now assume inductively that |S| + |S | ≥ 1, and the claim has already been proven for smaller values of |S| + |S |. Note that the case |S| = 0 is trivial. To establish the claim for |S| ≥ 1, it suffices to do so for the case |S| = 1. Indeed, if |S| ≥ 1, then we can split S = { j} ∪ (S\{ j}) for some j ∈ S. 32) then follows by applying the induction hypothesis to estimate the second factor. 32). Writing S = {i}, we reduce to showing that P Ai | A¯ j ≥ xi . j∈S We split S = S1 ∪ S2 where S1 := { j ∈ S|(i, j) ∈ E} are those indices j which are adjacent to i in the dependency graph, and S2 := S \S1 .

### Additive combinatorics by Terence Tao

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