Intersections of multiplicative translates of 3-adic Cantor sets

Motivated by a question of Erd\H{o}s, this paper considers questions concerning the discrete dynamical system on the 3-adic integers given by multiplication by 2. Let the 3-adic Cantor set consist of all 3-adic integers whose expansions use only the digits 0 and 1. The exception set is the set of 3-adic integers whose forward orbits under this action intersects the 3-adic Cantor set infinitely many times. It has been shown that this set has Hausdorff dimension 0. Approaches to upper bounds on the Hausdorff dimensions of these sets leads to study of intersections of multiplicative translates of Cantor sets by powers of 2. More generally, this paper studies the structure of finite intersections of general multiplicative translates of the 3-adic Cantor set by integers 1 < M_1 < M_2 < ...< M_n. These sets are describable as sets of 3-adic integers whose 3-adic expansions have one-sided symbolic dynamics given by a finite automaton. As a consequence, the Hausdorff dimension of such a set is always of the form log(\beta) for an algebraic integer \beta. This paper gives a method to determine the automaton for given data (M_1, ..., M_n). Experimental results indicate that the Hausdorff dimension of such sets depends in a very complicated way on the integers M_1,...,M_n.Comment: v1, 31 pages, 6 figure

On a conjecture of Wilf

Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum \sum_{j=0}^{n} (-1)^{j} S(n,j) is nonzero for all n>2. We prove this conjecture for all n not congruent to 2 and not congruent to 2944838 modulo 3145728 and discuss applications of this result to graph theory, multiplicative partition functions, and the irrationality of p-adic series.Comment: 18 pages, final version, accepted for publication in the Journal of Combinatorial Theory, Series

Feynman integrals and motives

This article gives an overview of recent results on the relation between quantum field theory and motives, with an emphasis on two different approaches: a "bottom-up" approach based on the algebraic geometry of varieties associated to Feynman graphs, and a "top-down" approach based on the comparison of the properties of associated categorical structures. This survey is mostly based on joint work of the author with Paolo Aluffi, along the lines of the first approach, and on previous work of the author with Alain Connes on the second approach.Comment: 32 pages LaTeX, 3 figures, to appear in the Proceedings of the 5th European Congress of Mathematic