By Terence Tao

Additive combinatorics is the idea of counting additive constructions in units. This conception has noticeable fascinating advancements and dramatic alterations in path in recent times because of its connections with parts resembling quantity idea, ergodic idea 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 assemble in a self-contained and systematic demeanour the numerous diverse instruments and ideas which are utilized in the fashionable idea, featuring them in an available, coherent, and intuitively transparent demeanour, and offering speedy purposes to difficulties in additive combinatorics. the facility of those instruments is easily confirmed within the presentation of modern advances comparable 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 a good number of routines and new effects.

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718 . . is the base of the natural logarithm. Let V be a finite set, and for each i ∈ V let Ai be a probabilistic event with P(Ai ) ≤ p. Assume also that each Ai is mutually independent of all but at most d of the other events A j . Then A¯ i P ≥ 1− i∈V 1 d +1 |V | > 0. 30). 25 by setting xi = d+1 and using the fact that (1 − d+1 ) > 1 . The constant e is best possible as shown by Shearer. 26. This is the original result from the paper [93] of Erd˝os and Lov´asz, which motivated the development of the local lemma.

Tn ) ≥ X (t1 , . . , tn ) whenever ti ≥ ti for all 1 ≤ i ≤ n or equivalently if X is monotone increasing in each of the variables ti separately. We call X monotone decreasing if −X is monotone increasing. We say that an event A is monotone increasing (resp. decreasing) if the indicator I(A) is monotone increasing (resp. decreasing). 18 If P(t1 , . . , tn ) is any polynomial of t1 , . . , tn with non-negative coefficients, then P is monotone increasing and −P is monotone decreasing, and the event P(t1 , .

16) is known as the exponential moment method. Of course, to use it effectively one needs to be able to compute the exponential moments E(et X ). 7 Let X be a random variable with |X | ≤ 1 and E(X ) = 0. Then for any −1 ≤ t ≤ 1 we have E(et X ) ≤ exp(t 2 Var(X )). Proof Since |t X | ≤ 1, a simple comparison of Taylor series gives the inequality et X ≤ 1 + t X + t 2 X 2 . Taking expectations of both sides and using linearity of expectation and the hypothesis E(X ) = 0 we obtain E(et X ) ≤ 1 + t 2 Var(X ) ≤ exp(t 2 Var(X )) as desired.