arXiv subjects

Rowan O'Clarey

Publications and source records attributed to Rowan O'Clarey.

3 recordsLinked to original sources

arXiv2026

On the Sum of a Prime and a Number that is not Square-Free

We prove that every sufficiently large integer $n$ can be written as the sum of a prime and an integer that is not square-free. In addition, we expect this result holds for every $n > 24$ and prove two results to support this claim. First, we prove the result holds unconditionally for every odd $n > 24$. Second, assuming the Generalised Riemann Hypothesis for Dirichlet $L$-functions, we prove the result holds for every $n > 24$. We also discuss the obstruction which prohibits us from proving the result unconditionally for every $n > 24$.

math.NT↗

arXiv2026

On the sum of a prime and a square-free number co-prime to any integer with at most two prime factors

Every natural number greater than $2$ can be written as the sum of a prime and a square-free number, and recent work has imposed additional divisibility conditions on the square-free number. We overcome limitations in these works to prove new results on square-free numbers co-prime to any integer with up to two prime factors, which make the expected asymptotic results explicit.

math.NT↗

arXiv2026

Chaos and Superconcentration for Poisson Functionals with Applications in Stochastic Geometry

We consider square-integrable functionals of Poisson point processes for which the variance upper bound provided by the classical Poincaré inequality is suboptimal, a phenomenon known as superconcentration. In this paper, we establish a rigorous mathematical equivalence between superconcentration and the chaotic behaviour of the functional, and certain associated random sets, under perturbations driven by the Ornstein-Uhlenbeck semigroup on the Poisson space. Leveraging the Malliavin-Stein method, we develop general variance identities and bounds for Poisson functionals, providing a unified framework to prove superconcentration, particularly for geometric functionals that can be expressed as a sum of local score functions. We apply our results to rigorously establish superconcentration and the chaotic behaviour in some models of stochastic geometry. Specifically, we analyse horizontal box-crossing indicators in certain critical continuum percolations, as well as the number of vertices with small degrees and the number of isolated $Γ$-components in random geometric graphs in the dense regime.

math.PR↗