arXiv subjects
Publications and source records attributed to Paul Balister.
In this paper we have shall generalize Shearer's entropy inequality and its recent extensions by Madiman and Tetali, and shall apply projection inequalities to deduce extensions of some of the inequalities concerning sums of sets of integers proved recently by Gyarmati, Matolcsi and Ruzsa. We shall also discuss projection and entropy inequalities and their connections.
Let P be a Poisson process of intensity one in a square S_n of area n. For a fixed integer k, join every point of P to its k nearest neighbours, creating an undirected random geometric graph G_{n,k}. We prove that there exists a critical constant c such that for c' c G_{n,c'\log n} is connected with probability tending to 1 as n tends to infinity. This answers a question previously posed by the authors.
Let A be the annulus in R^2 centered at the origin with inner and outer radii r(1-ε) and r, respectively. Place points {x_i} in R^2 according to a Poisson process with intensity 1 and let G_A be the random graph with vertex set {x_i} and edges x_ix_j whenever x_i-x_j\in A. We show that if the area of A is large, then G_A almost surely has an infinite component. Moreover, if we fix ε, increase r and let n_c=n_c(ε) be the area of A when this infinite component appears, then n_c\to1 as ε\to 0. This is in contrast to the case of a ``square'' annulus where we show that n_c is bounded away from 1.