By Terence Tao
Additive combinatorics is the idea of counting additive buildings in units. This concept has noticeable interesting advancements and dramatic alterations in course in recent times due to its connections with parts reminiscent of quantity idea, ergodic concept and graph concept. This graduate point textual content will permit scholars and researchers effortless access into this interesting box. the following, for the 1st time, the authors compile in a self-contained and systematic demeanour the various various instruments and concepts which are utilized in the fashionable idea, offering them in an available, coherent, and intuitively transparent demeanour, and delivering instant purposes to difficulties in additive combinatorics. the facility of those instruments is definitely established within the presentation of contemporary advances equivalent 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 means of a great number of workouts and new effects.
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Extra info for Additive combinatorics
N ) ∈ Zn+ , we define the partial derivative ∂ α Y as ∂ α Y := ∂ ∂t1 α1 ··· ∂ ∂tn αn Y (t1 , . . , 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 . 36 Let k ≥ 1, and let Y = Y (t1 , .
X n are jointly independent random variables where |X i − E(X i )| ≤ 1 for all i. Set X := X 1 + · · · + √ X n and let σ := Var(X ) be the standard deviation of X . Then for any λ > 0 P(|X − E(X )| ≥ λσ ) ≤ 2 max e−λ 2 /4 , e−λσ/2 . 17) asserts that X = E(X ) + O(Var(X ) ) with high probability, and X = E(X ) + O(ln1/2 nVar(X )1/2 ) with extremely high probability (1 − O(n −C ) for some large C). The bound in Chernoff’s theorem provides a huge improvement over Chebyshev’s inequality when λ is large.
Bk ∈ [0, N ], and conversely b1 , . . , bk ∈ [0, N ] implies n ≤ k N . In particular if B is a basis of order k then |B ∩ [0, N ]| = (N 1/k ). 22) Let us say that a basis B of order k is thin if rk,B (n) = O(log n) for all large n. 22). In the 1930s, Sidon asked the question of whether thin bases actually exist (or more generally, any basis which is “high quality” in the sense that rk,B (n) = n o(1) for all n). As Erd˝os recalled in one of his memoirs, he thought he could provide an answer within a few days.