Quantitative results using variants of Schmidt’s game: Dimension bounds, arithmetic progressions, and more

Ryan Broderick¹ ; Lior Fishman² ; David Simmons³

¹ Department of Mathematics University of California, Irvine Irvine, CA 92697-3875, U.S.A.
² Department of Mathematics University of North Texas Denton, TX 76203-5017, U.S.A. <a href="http://math.unt.edu/lior-fishman">math.unt.edu/lior-fishman</a>
³ Department of Mathematics University of York Heslington, York YO10 5DD, UK <a href="https://sites.google.com/site/davidsimmonsmath/">sites.google.com/site/davidsimmonsmath/</a>

Acta Arithmetica, Tome 188 (2019) no. 3, pp. 289-316.

Résumé

Schmidt’s game is generally used to deduce qualitative information about the Hausdorff dimensions of fractal sets and their intersections. However, one can also ask about quantitative versions of the properties of winning sets. In this paper we show that such quantitative information has applications to various questions including: $\bullet$ What is the maximal length of an arithmetic progression on the “middle $\epsilon$” Cantor set? $\bullet$ What is the smallest $n$ such that there is some element of the ternary Cantor set whose continued fraction partial quotients are all $\leq n$? $\bullet$ What is the Hausdorff dimension of the set of $\epsilon$-badly approximable numbers on the Cantor set? We show that a variant of Schmidt’s game known as the potential game is capable of providing better bounds on the answers to these questions than the classical Schmidt’s game. We also use the potential game to provide a new proof of an important lemma in the classical proof of the existence of Hall’s Ray.

Détail

DOI : 10.4064/aa171127-8-11

Keywords: schmidt game generally deduce qualitative information about hausdorff dimensions fractal sets their intersections however ask about quantitative versions properties winning sets paper quantitative information has applications various questions including bullet what maximal length arithmetic progression middle nbsp epsilon cantor set bullet what smallest there element ternary cantor set whose continued fraction partial quotients leq bullet what hausdorff dimension set epsilon badly approximable numbers cantor set variant schmidt game known potential game capable providing better bounds answers these questions classical schmidt game potential game provide proof important lemma classical proof existence hall ray