We construct minor-closed addable families of graphs that are subcritical and contain all planar graphs. This contradicts (one direction of) a well-known conjecture of Noy.

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We analyze the first-order asymptotic growth of a n = 1 0 n j=1 4 sin 2 (πjx)dx. The integer a n appears as the main term in a weighted average of the number of orbits in a particular quasihyperbolic automorphism of a 2n-torus, which has applications to ergodic and analytic number theory. The combinatorial structure of a n is also of interest, as the "signed" number of ways in which 0 can be represented as the sum of j j for −n ≤ j ≤ n (with j = 0), with j ∈ {0, 1}. Our result answers a question of Thomas Ward (no relation to the fourth author) and confirms a conjecture of Robert Israel and Steven Finch.Dedicated to the memory of Philippe Flajolet.

We study representations of integers as sums of the form ±a1 ± a2 ± · · · ± an, where a1, a2, . . . is a prescribed sequence of integers. Such a sequence is called an Erdős-Surányi sequence if every integer can be written in this form for some n ∈ N and choices of signs in infinitely many ways. We study the number of representations of a fixed integer, which can be written as a trigonometric integral, and obtain an asymptotic formula under a rather general scheme due to Roth and Szekeres. Our approach, which is based on Laplace's method for approximating integrals, can also be easily extended to find higher-order expansions. As a corollary, we settle a conjecture of Andrica and Ionaşcu on the number of solutions to the signum equation ±1 k ± 2 k ± · · · ± n k = 0.

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