arXiv subjects

Maiken Gravgaard

Publications and source records attributed to Maiken Gravgaard.

3 recordsLinked to original sources

arXiv2026

Quantitative Khintchine on the parabola with non-monotonic approximation functions

We prove a quantitative version of the convergence case of Khintchine's celebrated theorem in metric Diophantine approximation, but where the approximated points are restricted to lying on the parabola. A novel feature of our result is that unlike other results in literature, the approximating function is no longer required to be monotonic. This requires us to obtain explicit constants in classical number theoretic results, most notably in Burgess' bound for character sums in short intervals.

math.NT↗

arXiv2026

Lower bounds for the Hausdorff dimension of expressible sets

We obtain positive lower bounds on the Hausdorff dimension of sets of real numbers given by expressions of the form $\sum_{n=1}^\infty \frac{1}{a_n b_n}$, where $b_n$ satisfies some growth condition and $a_n$ lies in some set, possibly depending on $n$. As a consequence of our results, some of the irrational numbers arising from Erd\H{o}s' celebrated construction from 1976 are not Liouville numbers.

math.NT↗

arXiv2025

Decomposition of real numbers into sums of L\"uroth sets

We study the decomposition of real numbers into sums of L\"uroth sets, which are defined by numbers whose L\"uroth expansions have prescribed digit constraints. We establish several results on the congruence modulo 1 of sums of L\"uroth sets, including summands with digits bounded above, below, and combinations of the two. We also analyse the Hausdorff dimension of L\"uroth sets and their sums. The results extend classical findings on continued fractions to L\"uroth expansions.

math.NT↗