Maximal (k, l)-free sets in ℤ/pℤ are arithmetic progressions

Constructing Small Sets that are Uniform in Arithmetic Progressions

Maximal Arithmetic Progressions in Random Subsets

Let U (N) denote the maximal length of arithmetic progressions in a random uniform subset of {0, 1}N . By an application of the Chen-Stein method, we show that U −2 log N/ log 2 converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W (N) of arithmetic progressions (mod N). When considered in the natural way on a common probability space, we ...

Triangle Free Sets and Arithmetic Progressions - Two Pisier Type Problems

Let Pf (N ) be the set of finite nonempty subsets of N and for F,G ∈ Pf (N ) write F < G when maxF < minG. Let X = {(F,G) : F,G ∈ Pf (N ) and F < G}. A triangle in X is a set of the form {(F ∪ H,G), (F,G), (F,H ∪ G)} where F < H < G. Motivated by a question of Erdős, Nešetŕıl, and Rödl regarding three term arithmetic progressions, we show that any finite subset Y of X contains a relatively larg...

Product Sets of Arithmetic Progressions

In this paper, we generalize a result of Nathanson and Tenenbaum on sum and product sets, partially answering the problem raised at the end of their paper [N-T]. More precisely, they proved that if A is a large finite set of integers such that |2A| < 3|A| − 4, then |A2| > ( |A| `n |A| ) 2 |A|2−ε. It is shown here that if |2A| < α|A|, for some fixed α < 4, then |A2| |A|2−ε. Furthermore, if α < 3...

On the maximal length of arithmetic progressions∗

This paper is a continuation of Benjamini, Yadin and Zeitouni’s paper [4] on maximal arithmetic progressions in random subsets. In this paper the asymptotic distributions of the maximal arithmetic progressions and arithmetic progressions modulo n relative to an independent Bernoulli sequence with parameter p are given. The errors are estimated by using the Chen-Stein method. Then the almost sur...