We prove a uniform functional law of the logarithm for the local empirical process. To accomplish this we combine techniques from classical and abstract empirical process theory, Gaussian distributional approximation and probability on Banach spaces. The body of techniques we develop should prove useful to the study of the strong consistency of d-variate kernel-type nonparametric function estim...

Let p be a prime k|p−1, t = (p−1)/k and γ(k, p) be the minimal value of s such that every number is a sum of s k-th powers (mod p). We prove Heilbronn’s conjecture that γ(k, p) k1/2 for t > 2. More generally we show that for any positive integer q, γ(k, p) ≤ C(q)k1/q for φ(t) ≥ q. A comparable lower bound is also given. We also establish exact values for γ(k, p) when φ(t) = 2. For instance, whe...

Although the basic building blocks for working with strong encryption have become fairly widespread in the last few years, experience has shown that implementers frequently misuse them in a manner that voids their security properties. At least some of the blame lies with the tools themselves, which often make it unnecessarily easy to get things wrong. Just as no chainsaw manufacturer would thin...

ABSTRACT In this paper, we investigate the limitations of capacity-based random code constructions for the wiretap channel, i.e., constructions that associate to each confidential message a subcode whose rate approaches the capacity of the eavesdropper’s channel. Generalizing a previous result for binary symmetric channels, we show that random capacity-based codes do not achieve the strong secr...

We prove a version of Hrushovski's 1989 results on almost orthogonal regular types in the context of simple and superstable nitary abstract elementary classes: from a certain expression of `non-orthogonality' we can conclude the existence of a group acting on the geometry obtained on the set of realizations of a regular Lascar strong type, and if we rule out the presence of a non-classical grou...