Metric Diophantine Approximation on Fractals

Metric Diophantine Approximation on Fractals

Diophantine approximation

James Wyatt

2024

Named after Diophantus of Alexandria, Diophantine approximation is a field in number theory that studies how well real numbers can be approximated by rational numbers. In 1984 Mahler presented eight problems in transcendental number theory and Diophantine approximation. One of these problems posed the questions, ‘how well can irrationals in the middle third Cantor set be approximated by rationals in the middle third Cantor set?’ and ‘how well can irrationals in the middle third Cantor set be approximated by rationals not in the middle third Cantor set?’. This problem serves as the inspiration for this paper. The middle third Cantor set can be described as a missing digit set of base 3, which motivates our study of missing digit sets. In this paper, we will address the following related question, `How well can irrationals in a missing digit set be approximated by rationals with polynomial denominators?' and prove some related results. To achieve this, we will be closely looking at Khintchine's theorem, particularly the convergence case and aim to prove a Khintchine-like convergence theorem for missing digit sets with large bases and rationals with polynomial denominators.

tag1 mathematics, tag2 number theory, tag3 Diophantine approximation, tag4 measure theory, tag5 missing digit sets