By Terence Tao

Additive combinatorics is the idea of counting additive constructions in units. This idea has visible interesting advancements and dramatic alterations in path in recent times because of its connections with components similar to quantity thought, ergodic idea and graph conception. This graduate point textual content will permit scholars and researchers effortless access into this interesting box. right here, for the 1st time, the authors assemble in a self-contained and systematic demeanour the numerous varied instruments and concepts which are utilized in the trendy idea, providing them in an available, coherent, and intuitively transparent demeanour, and delivering speedy functions to difficulties in additive combinatorics. the facility of those instruments is easily proven 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 by way of loads of workouts and new effects.

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**Example text**

Do not contain any factor of ti2 ), and homogeneous if all the monomials have the same degree. Thus for instance a boolean polynomial is automatically regular and simplified, though not necessarily homogeneous. Given any multi-index α = (α1 , . . , αn ) ∈ 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 .

12 The squares N∧ 2 = {0, 1, 4, 9, . } are known to be a basis of order 4 (Legendre’s theorem), while the primes P = {2, 3, 5, 7, . } are conjectured to be a basis of order 3 (Goldbach’s conjecture) and are known to be a basis of order 4 (Vinogradov’s theorem). Furthermore, for any k ≥ 1, the kth powers N∧ k = {0k , 1k , 2k , . } are known to be a basis of order C(k) for some finite C(k) (Waring’s conjecture, first proven by Hilbert). 3 The exponential moment method 13 rm,N∧ k (n) = m,k (n k −1 ) for all large n, if m is sufficiently large depending on k (see for instance [379] for a discussion).

28. 28 We shall use the exponential moment method. By a limiting argument we may assume that P(t j = 0), P(t j = 1) > 0 for all j. We introduce the moment generating function F(t) := E(e−t X ) for any t > 0. 16) we have P(X ≤ E(X ) − T ) ≤ F(t) e−t(E(X )−T ) . 6 Janson’s inequality 31 Taking logarithms, we see that we only need to establish the inequality T2 2 for some t > 0. 8, the summands in X are not necessarily independent, so we cannot factorize F(t) = E(e−t X ) easily. Janson found a beautiful argument to get around this difficulty.