arXiv subjects

Andrew Pollington

Publications and source records attributed to Andrew Pollington.

5 recordsLinked to original sources

arXiv2025

General Duffin--Schaeffer-type counterexamples in diophantine approximation

Duffin and Schaeffer provided a famous counterexample to show that Khintchine's theorem fails without monotonicity assumption. Given any monotonically decreasing approximation function with divergent series, we construct Duffin--Schaeffer-type counterexamples by restricting the denominator. We also extend these constructions to the inhomogeneous setting. Our results resolve some natural questions arising from the works of Erd\H{o}s, Vaaler, and Yu.

math.NT↗

arXiv2024

Simultaneous visibility in the integer lattice

Two lattice points are visible from one another if there is no lattice point on the open line segment joining them. Let $S$ be a finite subset of $\mathbb{Z}^k$. The asymptotic density of the set of lattice points, visible from all points of $S$, was studied by several authors. Our main result is an improved upper bound on the error term. We also find the Schnirelmann density of the set of visible points from some sets S. Finally, we discuss these questions from the point of view of ergodic theory.

math.NT↗

arXiv2010

On a problem in simultaneous Diophantine approximation: Schmidt's conjecture

For any $i,j \ge 0$ with $i+j =1$, let $\bad(i,j)$ denote the set of points $(x,y) \in \R^2$ for which $ \max \{\|qx\|^{1/i}, \|qy\|^{1/j} \} > c/q $ for all $ q \in \N $. Here $c = c(x,y)$ is a positive constant. Our main result implies that any finite intersection of such sets has full dimension. This settles a conjecture of Wolfgang M. Schmidt in the theory of simultaneous Diophantine approximation.

math.NT↗

arXiv2009

The Duffin-Schaeffer Conjecture with extra divergence

Given a nonnegative function $ψ: \N \to \R $, let $W(ψ)$ denote the set of real numbers $x$ such that $|nx -a| < ψ(n) $ for infinitely many reduced rationals $a/n (n>0) $. A consequence of our main result is that $W(ψ)$ is of full Lebesgue measure if there exists an $ε> 0 $ such that $$ \textstyle \sum_{n\in\N}(\frac{ψ(n)}{n})^{1+ε}φ(n)=\infty . $$ The Duffin-Schaeffer Conjecture is the corresponding statement with $ε= 0$ and represents a fundamental unsolved problem in metric number theory. Another consequence is that $W(ψ)$ is of full Hausdorff dimension if the above sum with $ε= 0$ diverges; i.e. the dimension analogue of the Duffin-Schaeffer Conjecture is true.

math.NT↗

arXiv2005

Metric Diophantine approximation and 'absolutely friendly' measures

Let $W(\p)$ denote the set of $\p$-well approximable points in $\R^d$ and let $K$ be a compact subset of $\R^d$ which supports a measure $μ$. In this short note, we show that if $μ$ is an `absolutely friendly' measure and a certain $μ$--volume sum converges then $μ(W(\p) \cap K) = 0$. The result obtained is in some sense analogous to the convergence part of Khintchines classical theorem in the theory of metric Diophantine approximation. The class of absolutely friendly measures is a subclass of the friendly measures introduced by D. Kleinbock, E. Lindenstrauss and B. Weiss (On fractal measures and Diophantine approximation) and includes measures supported on self similar sets satisfying the open set condition. We also obtain an upper bound result for the Hausdorff dimension of $W(\p) \cap K $.

math.NT↗