Mei-Chu Chang

A polynomial bound in Freiman's theorem

.Earlier bounds involved exponential dependence in αin the second estimate. Ourargument combines I. Ruzsa’s method, which we improve in several places, as well asY. Bilu’s proof of Freiman’s theorem.A fundamental result in the theory of set addition is Freiman’s theorem. Let A ⊂Z be a finite set of integers with small sumset; thus assume|A + A| <α|A|, (0.1)whereA + A = {x + y |x,y ∈ A} (0.2)and | · | denotes the cardinality. The factor αshould be thought of as a (possiblylarge) constant. Then Freiman’s theorem states that A is contained in a d-dimensionalprogression P, whered ≤ d(α) (0.3)and|P||A|≤ C(α). (0.4)(Precise definitions are given in Section 1.) Although this statement is very intuitive,there is no simple proof so far, and it is one of the deep results in additive numbertheory.G. Freiman’s book [Fr] on the subject is not easy to read, which perhaps explainswhy in earlier years the result did not get its deserved publicity. More recently, two

On the size of -fold sum and product sets of integers

We prove the following theorem: for all positive integers

On problems of Erdös and Rudin

b

Additive and Multiplicative Structure in Matrix Spaces

there exists a positive integer

A sum-product estimate in algebraic division algebras

k

A Bound on the Order of Jumping Lines.

, such that for every finite set

An Estimate of Incomplete Mixed Character Sums

A

On sums and products of distinct numbers

of integers with cardinality

ELEMENTS OF LARGE ORDER IN PRIME FINITE FIELDS

|A| > 1

Double character sums over subgroups and intervals

, we have either