Finite field models in arithmetic combinatorics - ten years on
نویسنده
چکیده
منابع مشابه
Finite Field Models in Arithmetic Combinatorics
The study of many problems in additive combinatorics, such as Szemerédi’s theorem on arithmetic progressions, is made easier by first studying models for the problem in F p , for some fixed small prime p. We give a number of examples of finite field models of this type, which allows us to introduce some of the central ideas in additive combinatorics relatively cleanly. We also give an indicatio...
متن کاملAdditive Combinatorics and Theoretical Computer Science ∗ Luca Trevisan
Additive combinatorics is the branch of combinatorics where the objects of study are subsets of the integers or of other abelian groups, and one is interested in properties and patterns that can be expressed in terms of linear equations. More generally, arithmetic combinatorics deals with properties and patterns that can be expressed via additions and multiplications. In the past ten years, add...
متن کاملErgodic Methods in Additive Combinatorics
Shortly after Szemerédi’s proof that a set of positive upper density contains arbitrarily long arithmetic progressions, Furstenberg gave a new proof of this theorem using ergodic theory. This gave rise to the field of combinatorial ergodic theory, in which problems motivated by additive combinatorics are addressed kwith ergodic theory. Combinatorial ergodic theory has since produced combinatori...
متن کاملOn Growth in an Abstract Plane
There is a parallelism between growth in arithmetic combinatorics and growth in a geometric context. While, over R or C, geometric statements on growth often have geometric proofs, what little is known over finite fields rests on arithmetic proofs. We discuss strategies for geometric proofs of growth over finite fields, and show that growth can be defined and proven in an abstract projective pl...
متن کاملSum-Product and Character Sums in finite fields
In this talk I will present estimates on incomplete character sums in finite fields, with special emphasize on the non-prime case. Some of the results are of the same strength as Burgess celebrated theorem for prime fields. The improvements are mainly based on arguments from arithmetic combinatorics providing new bounds on multiplicative energy and an improved amplification strategy. In particu...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Finite Fields and Their Applications
دوره 32 شماره
صفحات -
تاریخ انتشار 2015