Dyadic approximation in the middle-third Cantor set

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Demi Allen
April 15, 2021
11:00AM - 11:55AM
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Add to Calendar 2021-04-15 11:00:00 2021-04-15 11:55:00 Dyadic approximation in the middle-third Cantor set Speaker:  Demi Allen (University of Bristol, UK) Title:   Dyadic approximation in the middle-third Cantor set Speaker's URL:  https://research-information.bris.ac.uk/en/persons/demi-d-allen Abstract:  Motivated by a classical question due to Mahler, in 2007 Levesley, Salp, and Velani showed that the Hausdorff measure of the set of points in the middle-third Cantor set which can be approximated by triadic rationals (that is, rationals which have denominators which are powers of 3) at a given rate of approximation satisfies a zero-full dichotomy. More precisely, the Hausdorff measure of the set in question is either zero or full according to, respectively, the convergence or divergence of a certain sum which is dependent on the specified rate of approximation. Naturally, one might also wonder what can be said about dyadic approximation in the middle-third Cantor set. That is, how well can we approximate points in the middle-third Cantor set by rationals which have denominators which are powers of 2? In this talk I will discuss a conjecture on this topic due to Velani, some progress towards this conjecture, and why dyadic approximation is harder than triadic approximation in the middle-third Cantor set. This talk will be based on joint work with Sam Chow (Warwick) and Han Yu (Cambridge). Zoom:  https://osu.zoom.us/j/95531240801?pwd=M3VjMUtRUDAwUmpzV3hnSVIzVnU1QT09 Meeting ID: 955 3124 0801 Password: Analysis Zoom info below Department of Mathematics math@osu.edu America/New_York public
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Speaker:  Demi Allen (University of Bristol, UK)

Title:   Dyadic approximation in the middle-third Cantor set

Speaker's URL:  https://research-information.bris.ac.uk/en/persons/demi-d-allen

Abstract:  Motivated by a classical question due to Mahler, in 2007 Levesley, Salp, and Velani showed that the Hausdorff measure of the set of points in the middle-third Cantor set which can be approximated by triadic rationals (that is, rationals which have denominators which are powers of 3) at a given rate of approximation satisfies a zero-full dichotomy. More precisely, the Hausdorff measure of the set in question is either zero or full according to, respectively, the convergence or divergence of a certain sum which is dependent on the specified rate of approximation. Naturally, one might also wonder what can be said about dyadic approximation in the middle-third Cantor set. That is, how well can we approximate points in the middle-third Cantor set by rationals which have denominators which are powers of 2? In this talk I will discuss a conjecture on this topic due to Velani, some progress towards this conjecture, and why dyadic approximation is harder than triadic approximation in the middle-third Cantor set. This talk will be based on joint work with Sam Chow (Warwick) and Han Yu (Cambridge).

Zoom:  https://osu.zoom.us/j/95531240801?pwd=M3VjMUtRUDAwUmpzV3hnSVIzVnU1QT09

Meeting ID: 955 3124 0801

Password: Analysis

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