By Terence Tao

Additive combinatorics is the speculation of counting additive buildings in units. This concept has noticeable fascinating advancements and dramatic adjustments in course in recent times due to its connections with components resembling quantity thought, ergodic idea and graph conception. This graduate point textual content will enable 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 various instruments and ideas which are utilized in the trendy conception, offering them in an obtainable, coherent, and intuitively transparent demeanour, and delivering instant purposes to difficulties in additive combinatorics. the facility of those instruments is definitely verified within the presentation of contemporary advances reminiscent of 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 plenty of workouts and new effects.

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3. 15, which asserted for each k ≥ 1 the existence of a base B of order k with rk,B (n) = Ok (log n) for all large n. 13) using Chernoff’s inequality, but that method does not directly apply for higher k because rk,B (n) cannot be easily expressed as the sum of independent random variables. 1 The probabilistic method 38 We begin with a simple lemma on boolean polynomials that shows that if E(X ) is not too large, then at most points (t1 , . . , tn ) of the sample space, the polynomial X does not contain too many independent terms (cf.

19) 1 The probabilistic method 12 Applying this with = 1/2 (for instance), we conclude in particular that P(X = (E(X ))) ≥ 1 − 2e−E(X )/16 . 1) we have that |ti − E(ti )| ≤ 1 and Var(ti ) ≤ E(ti ). 9), we conclude that Var(X ) ≤ E(X ) (cf. 12)). 8 with λ := E(X )/σ . 4) we obtain the following concentration of measure property for the distribution of certain types of random sets. 10 Let A be a set (possibly infinite), and let B ⊂ A be a random subset of A with the property that the events a ∈ B are independent for every a ∈ A.

8 cannot be improved except for the constant in the exponent. 8, but with the X i complex-valued instead of real-valued. Show that E(|X − E(X )| ≥ λσ ) ≤ 4 max e−λ 2 /8 for all λ > 0. ) 2 √ , eλσ/2 2 √1 λσ 2 or |Im(z)| ≥ The constants here can be improved slightly. (Hoeffding’s inequality) Let X 1 , . . , X n be jointly independent random variables, taking finitely many values, with ai ≤ X i ≤ bi for all i and some real numbers ai < bi . Let X := X 1 + · · · + X n . Using the exponential moment method, show that ⎛ ⎞ P ⎝|X − E(X )| ≥ λ 1/2 n |bi − ai |2 ⎠ ≤ 2e−2λ2 .