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David Treeby

Publications and source records attributed to David Treeby.

5 recordsLinked to original sources
arXiv2026

Dense signed sums of non-integer powers

We prove that if $j>0$ is not an integer, then there is a choice of signs $\varepsilon_k\in\{\pm1\}$ such that the partial sums $ \sum_{k=1}^{N}\varepsilon_k k^j $ are dense in $\mathbb R$. The proof groups consecutive terms into Thue--Morse blocks, whose Prouhet--Tarry--Escott cancellation produces nonzero block sums tending to zero but with divergent total variation. A standard steering argument then chooses block signs so that the resulting partial sums visit arbitrarily small neighbourhoods of every real number.

math.GM↗
arXiv2026

Structure and Growth of Galileo Sequences

A Galileo sequence \((a_n)\) is a sequence of positive integers whose partial sums $S_n$ satisfy $S_{2n}=kS_n$ for some $k>1$. In this paper we prove that every polynomial Galileo sequence is given by first differences of the form \(a_n= C\left(n^d-(n-1)^d\right)\). We then show that every positive Galileo sequence has a binary-tree representation. Finally, for positive monotone integer-valued Galileo sequences, we prove power-law growth bounds, and give a continuous analog together with a characterization of all continuous solutions.

math.GM↗
arXiv2026

Straight-line optimality in Bellman's lost-in-a-forest problem for Euclidean balls

We prove that among all unit-speed paths, a straight line minimises the expected escape time from a ball in $\mathbf{R}^n$, solving the min-mean variant of Bellman's Lost~in~a~Forest problem for ball-shaped forests. The proof uses the Kneser--Poulsen conjecture in the plane, together with results on polygonal chain straightening in higher dimensions. Moreover, we calculate this minimal escape time by deriving the expected linear distance to the boundary of a ball in $n$ dimensions.

math.PR↗
arXiv2025

Pick-up Sticks and the Fibonacci Factorial

We present a variation of the broken stick problem in which $n$ stick lengths are sampled uniformly at random. We prove that the probability that no three sticks can form a triangle is the reciprocal of the product of the first $n$ Fibonacci numbers. Extensions to quadrilaterals and general $k$-gons are also discussed.

math.PR↗