Rational approximation with chosen numerators

TL;DR

The paper investigates how well real numbers can be approximated by rationals with prime denominators and a single fixed numerator per denominator, introducing the sets $\\mathscr{A}(\\bm{a},c)$ and $\\mathscr{A}(\\bm{a})$ and analyzing both probabilistic (random $\\bm{a}$) and deterministic constructions. It proves that for almost all sequences $\\bm{a}$, the full-measure property $\\lambda(\\mathscr{A}(\\bm{a}))=1$ holds, and constructs explicit deterministic sequences (a greedy sequence and sequences tied to Diophantine properties of $\\beta$) with the same feature; these results connect to twisted Diophantine approximation and yield Hausdorff-dimension statements for exceptional sets. The work also demonstrates applications to trace-function ergodic averages, showing convergence along sparse subsequences and non-convergence along all primes in certain regimes, and it develops a sieve-analytic perspective on the coverage problem. The combination of probabilistic, deterministic, and Diophantine-approximation methods, together with the Cantor-set approach to Hausdorff dimensions, provides a comprehensive view of how numerical representations with restricted denominators interact with distributional and dimension-theoretic properties. The findings have implications for equidistribution, trace-function dynamics, and the structure of exceptional sets in metric Diophantine problems.

Abstract

We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We obtain basic results, both probabilistic and deterministic, draw connections to twisted diophantine approximation, and present a simple application, related to possible correlations between trace functions and dynamical sequences.

Theorems & Definitions (51)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Remark 1.10