We show that with high probability a random subset of {1, . . . , n} of size Θ(n) contains two elements a and a + d, where d is a positive integer. As a consequence, we prove an analogue of the Sárközy-Fürstenberg theorem for a random subset of {1, . . . , n}.

We explore further the hardness of the non-dual discrete variant of the Ring-LWE problem for various number rings, give improved attacks for certain rings satisfying some additional assumptions, construct a new family of vulnerable Galois number fields, and apply some number theoretic results on Gauss sums to deduce the likely failure of these attacks for 2-power cyclotomic rings and unramified...

Abstract. Let H be a permutation group on a set Λ, which is permutationally isomorphic to a finite alternating or symmetric group An or Sn acting on the k-element subsets of points from {1, . . . , n}, for some arbitrary but fixed k. Suppose moreover that no isomorphism with this action is known. We show that key elements of H needed to construct such an isomorphism φ, such as those whose image...

In a complete financial market every contingent claim can be hedged perfectly. In an incomplete market it is possible to stay on the safe side by superhedging. But such strategies may require a large amount of initial capital. Here we study the question what an investor can do who is unwilling to spend that much, and who is ready to use a hedging strategy which succeeds with high probability.

In this survey we describe the Learning with Errors (LWE) problem, discuss its properties, its hardness, and its cryptographic applications.

We show that several versions of Floyd and Rivest’s algorithm Select for finding the kth smallest of n elements require at most n+min{k, n− k}+ o(n) comparisons on average and with high probability. This rectifies the analysis of Floyd and Rivest, and extends it to the case of nondistinct elements. Our computational results confirm that Select may be the best algorithm in practice.

Suppose we sequentially throw m balls into n bins. It is a natural question to ask for the maximum number of balls in any bin. In this paper we shall derive sharp upper and lower bounds which are reached with high probability. We prove bounds for all values of m(n) ≥ n/polylog(n) by using the simple and well-known method of the first and second moment.