On the Union of Arithmetic Progressions

We show that for every $\varepsilon>0$ there is an absolute constant $c(\varepsilon)>0$ such that the following is true. The union of any $n$ arithmetic progressions, each of length $n$, with pairwise distinct differences must consist of at least $c(\varepsilon)n^{2-\varepsilon}$ elements. We observe, by construction, that one can find $n$ arithmetic progressions, each of length $n$, with pairwise distinct differences such that the cardinality of their union is $o(n^2)$. We refer also to the non-symmetric case of $n$ arithmetic progressions, each of length $\ell$, for various regimes of $n$ and $\ell$

The Dynamical Mordell-Lang problem

Let X be a Noetherian space, let f be a continuous self-map on X, let Y be a closed subset of X, and let x be a point on X. We show that the set S consisting of all nonnegative integers n such that f^n(x) is in Y is a union of at most finitely many arithmetic progressions along with a set of Banach density zero. In particular, we obtain that given any quasi-projective variety X, any rational self-map map f on X, any subvariety Y of X, and any point x in X whose orbit under f is in the domain of definition for f, the set S is a finite union of arithmetic progressions together with a set of Banach density zero. We prove a similar result for the backward orbit of a point

New bounds for Szemeredi's theorem, II: A new bound for r_4(N)

Define r_4(N) to be the largest cardinality of a set A in {1,...,N} which does not contain four elements in arithmetic progression. In 1998 Gowers proved that r_4(N) 0. In this paper (part II of a series) we improve this to r_4(N) << N e^{-c\sqrt{log log N}}. In part III of the series we will use a more elaborate argument to improve this to r_4(N) << N(log N)^{-c}.Comment: 26 page

Yet another proof of Szemeredi's theorem

Using the density-increment strategy of Roth and Gowers, we derive Szemeredi's theorem on arithmetic progressions from the inverse conjectures GI(s) for the Gowers norms, recently established by the authors and Ziegler.Comment: 6 page note, to appear in the proceedings of a conference in honour of the 70th birthday of Endre Szemered