By Terence Tao

ISBN-10: 0521853869

ISBN-13: 9780521853866

Additive combinatorics is the speculation of counting additive constructions in units. This concept has obvious fascinating advancements and dramatic alterations in path in recent times due to its connections with components comparable to quantity thought, ergodic thought 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 collect in a self-contained and systematic demeanour the numerous assorted instruments and concepts which are utilized in the fashionable idea, providing them in an available, coherent, and intuitively transparent demeanour, and supplying quick functions to difficulties in additive combinatorics. the ability 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 through a number of routines and new effects.

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**Sample text**

40) E(n) := {(x1 , . . 1 }. R(n) + k 2 E(n). We view R(n) as the main term and E(n) as the error term; this reflects the intuitive fact that for most representations n = x1 + · · · + xk , the xi will be distinct and comparable in magnitude to n. It will suffice to show that with probability 1 we have E(n) = OC,k,B (1); R(n) = C,k,B (log n) for all but finitely many n. Let us deal first with the error term E(n). 43. 1 . 8 Thin bases of higher order 41 where Dn is the largest number of disjoint sets that one can find in An .

For r ≥ 3, much less was known. In 1980, Nathanson [259] proved that N∧r contains a subbasis of some order with density o(N 1/r ). 47, when A = N∧r .

15, proved above, asserts that N contains a thin basis of any order. 46 Let A be any fixed basis of order k. Does A contain a thin subbasis B? Note that Sidon’s original question can be viewed as the k = 2, A = N case of this question. 21) we know that a thin basis B enjoys the bounds |B ∩ [0, N ]| = k N 1/k ; |B ∩ [0, N ]| = Ok N 1/k log1/k N for all large N . 47 Let A be any fixed basis of order k. Does A contain a subbasis B with |B ∩ [0, N ]| = Ok (N 1/k log1/k N ) for all large N ? 47 has been investigated intensively for the Waring bases N∧r = {0 , 1r , 2r , .

### Additive combinatorics by Terence Tao

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