Talk Abstracts

Abstract: Diophantine Approximation is a branch of Number Theory in which the central theme is understanding how well real numbers can be approximated by rationals. In the most classical setting, a $\psi$-well-approximable number is one which can be approximated by rationals to a given degree of accuracy specified by an approximating function $\psi$. Khintchine's Theorem provides a beautiful characterisation of the Lebesgue measure of the set of $\psi$-well-approximable numbers and is one of the cornerstone results of Diophantine Approximation.

In this talk I will discuss the generalisation of Khintchine's Theorem to the setting of approximation for systems of linear forms. More specifically, I aim to discuss inhomogeneous Diophantine approximation for systems of linear forms subject to certain primitivity constraints. In one direction, we answer questions posed in this area by Dani, Laurent, and Nogueira (Mathematische Zeitschrift, 2015). In another, slightly different direction, we prove a univariate inhomogeneous version of the Duffin-Schaeffer Conjecture for systems of linear forms in at least three variables. This talk will be based on joint work with Felipe Ramírez (Wesleyan, US). 

Abstract: In this talk I will discuss some recent results on quantitative recurrence and the shrinking target problem for dynamical systems coming from overlapping iterated function systems. Such iterated function systems have the important property that a point often has several distinct choices of forward orbit. This non-uniqueness leads to different behaviour to that observed in the traditional setting where every point has a unique forward orbit. This talk will be based upon joint work with Henna Koivusalo (Bristol). 

Abstract: There are many self-similar measures whose Fourier transform does not tend to 0 as the frequency increases to infinity. Nonetheless, in joint work with Simon Baker, we show that given any non-atomic self-similar measure on the line, its pushforward under any C^2 map whose second derivative is everywhere positive has polynomial Fourier decay.

Abstract: We state a theorem on the existence of an interval in the parameter space for Okamoto's functions for which the graph always admits a horizontal slice with three elements. We show how this is analogous to a result of Sidorov's on base q expansions.  

Abstract: I will discuss some problems and results on inhomogeneous Diophantine approximation arising from Khintchine's theorem. In particular, I will discuss the Borel-Cantelli Lemma and zero-one laws in the context of the Inhomogeneous version of the Duffin-Schaeffer Conjecture and touch upon results and problems in higher dimensions. 

Abstract:  In this talk, I will discuss some combinatorial conditions that are imposed on the base-p expansion of any counterexample to the p-adic Littlewood Conjecture (pLC) of de Mathan and Teulié . Notably, I will show that for each fixed d>0 the length of the (2+d)-powers that appear in the base-p expansion of a counterexample to pLC are bounded. This is based on joint work with S. Kristensen and M.J. Northey. 

Abstract: The fractal dimension of state space sets is typically estimated via the scaling of either the generalized (R\'enyi) entropy or the correlation sum versus a size parameter, and more recently, via a new method based on extreme value theory.

Motivated by the lack of quantitative and systematic comparisons of fractal dimension estimators in the literature, and also by new methods both for estimating fractal dimensions and for performing delay embeddings, in this paper we provide a detailed and quantitative comparison for estimating the fractal dimension.

We start with summarizing existing estimators and then perform an evaluation of these estimators, comparing their performance and precision using different data sets and taking into account the impact of features like length, noise, embedding dimension, non-stationarity, falsify-ability, among many others.

Our analysis shows that for synthetic data the correlation sum and extreme value theory perform equally well as best estimators. Our results also challenge the widely established notion that the correlation sum method cannot be applied to high dimensional data. For real experimental data the entropy is better than the correlation sum, but the extreme value theory is equally good to the entropy. It appears that overall the more recent extreme value theory estimator is very powerful, but it also has two downsides when it comes to real data. First, it over-estimates the dimension values for short data lengths, and second, it will also yield low-dimensional deterministic results even for inappropriate data like stock market timeseries, requiring caution when applied to data of unknown underlying dynamics.

All quantities discussed are implemented as performant and easy to use open source code via the DynamicalSystems.jl library.

Abstract: We will consider a number of questions of the form: when does a plane measurable set realise all small or large distances, or alternatively contain small or large similar copies of a given finite set? The talk includes joint work with Vjekoslav Kovač, John Marstrand and Alexia Yavicoli. 

Abstract: I will briefly explain the realization of topological factoring between two zero dimensional systems by ordered "ordered premorphisms" between their associated Bratteli diagrams.

Abstract: We give a brief overview of an extension to Mahbub Alam and Anish Ghosh's generalisation of Dirichlet's Theorem, concerning equidistribution of rational approximates in a fixed number field. The technique uses ergodic theory and a generalisation of Rogers' Mean Value Theorem to derive an error term for the count of these rational approximates in any given direction.

Abstract: The shrinking target problem involves looking at  a dynamical system and the set of orbits which hit some shrinking sequence of sets infinitely often. Depending on the setting natural problems to consider are the measure of the set and the Hausdorff dimension. In the non-overlapping self-similar setting in $\R^d$ these problems are well understood when the sets are balls. In this talk we consider a natural family of self-similar sets including the Sierpinski gasket and carpet and the case when the shrinking targets are rectangles. We will show how in this case the Hausdorff dimension of the shrinking target set does not just depend on the size of the rectangle. This is joint work with Demi Allen and Benjamin Ward. 

Abstract: The first mass transference principle (MTP) was proved by Beresnevich and Velani in 2005. Roughly speaking, MTP says that if the limsup set given by a sequence of balls is large (full measure), shrinking the balls in a controlled way does not reduce the size of the limsup set too much (the dimension can be bound from below). MTP and its variants are an excellent tool for finding lower bounds for limsup-type sets. It has found many applications, including in Diophantine approximation, where it has been used to study well approximable points: Those points which lie in the limsup set of neighbourhoods of rational points.

We provide a new technique which relies on MTP but allows for the study of a particular liminf set: the set of badly approximable points. These are the points which eventually lie outside of the neighbourhoods of rationals. In particular, we show that the set of \tau-badly approximable points as a subset of \tau-well approximable points has full dimension. The special case of ambient dimension 1 was studied by Bugeaud, but his proof technique relies on continued fractions and does not generalise to higher dimensions.

The talk is based on a joint work with Ben Ward, Jason Levesley and Xintian Zhang.

Abstract: The purpose of this talk is to present a “possible new branch of ergodic theory” and ask whether it is potentially interesting or useful. Namely, for a self-map of a measurable space, I consider those probability measures that agree with their image measure on a pre-specified non-invariant sub-sigma-algebra. I will present an initial result on this topic. My original motivation was the development of an “entirely abstract” formalism for stationary measures of stationary-coloured-noise-driven dynamics, but this idea of “restrictedly invariant” measures seems interesting in its own right. 

Abstract: A pdf of the abstract can be viewed here.

Abstract: I will discuss an approach for computing the Hausdorff dimension of an intersection of the classical Lagrange and Markov spectra with half-infinite ray d(t) = dim(M \cap (−∞,t)), that allows to plot a graph of the function d(t) with high accuracy. The talk is based on a recent joint work with Carlos Gustavo Moreira and Carlos Matheus Santos (arxiv:2212.11371). 

Abstract: Khinchin's theorem is a nice result that tells us how we can approximate a Lebesgue generic point by rational points. It is curious to see what holds still if we replaced the Lebesgue measure with another Borel probability measure. Should we expect a significant change? Heuristically, unless the Borel probability measure is 'specially aligned' with rational points, we should expect that the statement of Khinchin's theorem holds without any change other than the replacement of the underlying measure. Simple examples of such measures are surface measures on 'curved manifolds', irrational affine subspaces, and some natural fractal measures like the middle third Cantor measure. In this talk, I will focus on the case with fractal measures and survey some recent partial progresses toward proving Khinchin's theorem on the middle 1/3 Cantor set. Part of this talk is based on the ongoing project with Demi Allen, Sam Chow, and Peter Varju.