Abstract. We prove that every 3-regular, n-vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n.Our method uses invariant Gaussian processes on the d-regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue λ. We show that such processes can be approximated by i.i.d. factors provided that |λ| ≤ 2 √ d − 1. We then use these approximations for λ = −2 √ d − 1 to produce factor of i.i.d. independent sets on regular trees.

A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum λmin of the spectrum of the adjacency matrix. To complement this result we use random eigenvectors to gain lower bounds in the vertex-transitive case. For example, we prove that the independence ratio of a 3-regular transitive graph is at leastThe same bound holds for infinite transitive graphs: we construct factor of i.i.d. independent sets for which the probability that any given vertex is in the set is at least q − o(1).We also show that the set of the distributions of factor of i.i.d. processes is not closed w.r.t. the weak topology provided that the spectrum of the graph is uncountable.

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We study factor of i.i.d. processes on the d-regular tree for d 3. We show that if such a process is restricted to two distant connected subgraphs of the tree, then the two parts are basically uncorrelated. More precisely, any functions of the two parts have correlation at most k(d − 1)/( √ d − 1) k , where k denotes the distance between the subgraphs. This result can be considered as a quantitative version of the fact that factor of i.i.d. processes have trivial 1-ended tails.

Abstract. Consider a 1 , . . . , a n ∈ R arbitrary elements. We characterize those functions f : R → R that decompose into the sum of a j -periodic functions, i.e., f = f 1 +· · ·+f n with ∆ a j f (x) := f (x+a j )−f (x) = 0. We show that f has such a decomposition if and only if for all partitions B 1 ∪B 2 ∪· · ·∪B N = {a 1 , . . . , a n } with B j consisting of commensurable elements with least common multiplesActually, we prove a more general result for periodic decompositions of functions f : A → R defined on an Abelian group A; in fact, we even consider invariant decompositions of functions f : A → R with respect to commuting, invertible self-mappings of some abstract set A.We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real-valued periodic decomposition of an integer-valued function implies the existence of an integer-valued periodic decomposition with the same periods.

A finite set H in R d is called an acute set if any angle determined by three points of H is acute. We examine the maximal cardinality αðdÞ of a d-dimensional acute set. The exact value of αðdÞ is known only for d ≤ 3. For each d ≥ 4 we improve on the best known lower bound for αðdÞ. We present different approaches. On one hand, we give a probabilistic proof that αðdÞ > c · 1:2 d . (This improves a random construction given by Erdős and Füredi.) On the other hand, we give an almost exponential constructive example which outdoes the random construction in low dimension (d ≤ 250). Both approaches use the small dimensional examples that we found partly by hand (d ¼ 4, 5) and partly by computer (6 ≤ d ≤ 10). We also investigate the following variant of the above problem: what is the maximal size κðdÞ of a d-dimensional cubic acute set (that is, an acute set contained in the vertex set of a d-dimensional hypercube)? We give an almost exponential constructive lower bound, and we improve on the best known upper bound.

We present a simple construction of an acute set of size 2 d−1 + 1 in R d for any dimension d. That is, we explicitly give 2 d−1 + 1 points in the d-dimensional Euclidean space with the property that any three points form an acute triangle. It is known that the maximal number of such points is less than 2 d . Our result significantly improves upon a recent construction, due to Dmitriy Zakharov, with size of order ϕ d where ϕ = (1 + √ 5)/2 ≈ 1.618 is the golden ratio.2010 Mathematics Subject Classification. 51M04, 51M15.

This paper is concerned with factors of independent and identically distributed processes on the $d$ -regular tree for $d\geq 3$ . We study the mutual information of values on two given vertices. If the vertices are neighbors (i.e. their distance is $1$ ), then a known inequality between the entropy of a vertex and the entropy of an edge provides an upper bound for the (normalized) mutual information. In this paper we obtain upper bounds for vertices at an arbitrary distance $k$ , of order $(d-1)^{-k/2}$ . Although these bounds are sharp, we also show that an interesting phenomenon occurs here: for any fixed process, the rate of mutual information decay is much faster, essentially of order $(d-1)^{-k}$ .