A quantitative improvement for Roth's theorem on arithmetic progressions
نویسنده
چکیده
We improve the quantitative estimate for Roth’s theorem on threeterm arithmetic progressions, showing that if A ⊂ {1, . . . , N} contains no nontrivial three-term arithmetic progressions then |A| N(log logN)4/ logN . By the same method we also improve the bounds in the analogous problem over Fq [t] and for the problem of finding long arithmetic progressions in a sumset.
منابع مشابه
Analytic and Combinatorial Number
These are lecture notes for the summer semester 2008 of the course Analytic and combinatorial number theory (NDMI045, Analytická a kombinatorická teoriě císel) which I have been teaching on the Faculty of Mathematics and Physics of the Charles University in Prague. The first booklet (the second one, [25], is for summer semester 2010) covers four major theorems, three on the distribution of prim...
متن کاملRoth’s Theorem on Arithmetic Progressions
The purpose of this paper is to provide a simple and self-contained exposition of the celebrated Roth's theorem on arithmetic progressions of length three. The original result is proved in [Roth53], while the proof given below is very similar to the exposition of Roth's original argument given in [GRS1990]. Definition. We say that a subset of positive integers A has positive upper density if (1...
متن کاملSelf-similar fractals and arithmetic dynamics
The concept of self-similarity on subsets of algebraic varieties is defined by considering algebraic endomorphisms of the variety as `similarity' maps. Self-similar fractals are subsets of algebraic varieties which can be written as a finite and disjoint union of `similar' copies. Fractals provide a framework in which, one can unite some results and conjectures in Diophantine g...
متن کاملOn rainbow 4-term arithmetic progressions
{sl Let $[n]={1,dots, n}$ be colored in $k$ colors. A rainbow AP$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. Conlon, Jungi'{c} and Radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow AP(4) free, when $n$ is even. Based on their construction, we show that such a coloring of $[4n]$...
متن کاملDepartment of Computer Science Spring 2008 9 Course 236610 : Research Laboratory in Theoretical Computer Science
Preface In our research laboratory we plan to study selected results from the eld of additive com-binatorics and some of their applications to theoretical computer science. A tentative list of results includes • Roth's Theorem about the existence of arithmetic progressions of length 3 in dense subsets of Z m p. • The Gowers norm and Samorodnitsky's Theorem about testing quadratic functions over...
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:
- J. London Math. Society
دوره 93 شماره
صفحات -
تاریخ انتشار 2016