By Terence Tao
Additive combinatorics is the idea of counting additive buildings in units. This concept has visible intriguing advancements and dramatic alterations in course in recent times due to its connections with parts resembling quantity concept, ergodic conception and graph thought. This graduate point textual content will let 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 numerous diverse instruments and concepts which are utilized in the trendy concept, proposing them in an available, coherent, and intuitively transparent demeanour, and supplying rapid purposes to difficulties in additive combinatorics. the facility of those instruments is definitely established within the presentation of modern 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 through plenty of routines and new effects.
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Let Y be the number of triangles in G(n, p). Give an upper bound and a lower bound for 3 P Y ≥ E(Y ) . 39. 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 , .
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.
39. 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.
Additive combinatorics by Terence Tao