By Terence Tao

ISBN-10: 0521853869

ISBN-13: 9780521853866

Additive combinatorics is the speculation of counting additive buildings in units. This concept has noticeable fascinating advancements and dramatic alterations in path lately because of its connections with parts comparable to quantity thought, ergodic conception and graph idea. This graduate point textual content will let scholars and researchers effortless access into this attention-grabbing box. the following, for the 1st time, the authors compile in a self-contained and systematic demeanour the numerous varied instruments and concepts which are utilized in the fashionable concept, featuring them in an available, coherent, and intuitively transparent demeanour, and supplying instant functions to difficulties in additive combinatorics. the facility of those instruments is easily established 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 via lots of workouts and new effects.

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**Sample text**

Tn ) of the sample space, the polynomial X does not contain too many independent terms (cf. 12). 40 Let X = A∈A j∈A t j be a boolean polynomial of n independent boolean variables t1 , . . , tn , let B ⊆ [1, n] be the random set B := { j ∈ [1, n] : t j = 1}, and let D ∈ N be the random variable, defined as the largest number of disjoint sets in A which are contained in B. Then for any integer K ≥ 1 we have E(X ) K . K! Observe that for A1 , . . , Ak disjoint, P(D ≥ K ) ≤ Proof I(D ≥ K ) ≤ 1 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 ), . . , Ek−1 (Y ) ≤ n −γ for all large n and some γ > 0. In other words, we need to establish E ∂ ∂t1 α1 ... ∂ ∂tn αn Y (t1 , . . , tn ) ≤ n −γ 1 The probabilistic method 42 whenever n is large and 1 ≤ α1 + · · · + αn ≤ k − 1.

By Bayes’ formula we thus have E(X Y ) = E(X Y |tn = 0)P(tn = 0) + E(X Y |tn = 1)P(tn = 1) ≥ E(X |tn = 1)E(Y |tn = 1)P(tn = 1). 25) and another application of the total probability formula we have E(X )E(Y ) = E(X |tn = 1)P(tn = 1)E(Y |tn = 1)P(tn = 1). 26). 20 Let A and B be two increasing events, then P(A ∧ B) ≥ P(A)P(B). 4 Correlation inequalities 21 More generally, if A1 , . . , Ak are increasing events, then P(A1 ∧ · · · ∧ Ak ) ≥ P(A1 ) · · · P(Ak ). 1 Asymptotic complementary bases Now we are going to use the FKG inequality to prove a result of Ruzsa [293] concerning asymptotic complementary bases.

### Additive combinatorics by Terence Tao

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