Good Distribution and Bad Approximability – GoDiBAp

Diophantine Approximation (DA) is a branch of Number Theory dating back to antiquity that can loosely be described as a quantitative analysis of the property that every real number can be approximated by rationals arbitrarily closely. Today, the theory is deeply intertwined with many areas of mathematics such as ergodic theory, probability theory and fractal geometry.

The many interactions between DA and other disciplines can be explained by the universal need to approximate complex structures by more regular ones. In this respect, a pivotal role is played by the set of badly approximable points : these are vectors admitting the worst possible rational approximants (in a suitable sense) but enjoying optimal distribution properties.

The proposed research aims to extend the boundaries in the current theory of bad approximability with a special focus on its multiplicative aspect centered around the Littlewood conjecture (LC). It is structured around three main objectives :

(A) Unifying the Additive and Multiplicative Theories of Bad approximability : the objective is to solve a fundamental problem due to Bugeaud to realise LC as a problem of interpolation with the well-understood set of additively badly approximable vectors;

(B) Developing a Geometric Theory of Bad Approximability on Translation Surfaces : characterising bad approximability as a property of equidistribution over the torus, the goal is to elaborate a far-reaching extension of this property over so-called translation surfaces. LC then appears as a special case of a family of geometric problems ;

(C) Developing a Structural Theory of Number Walls to solve P(t)–LC : a function field analogue of the property of bad approximability is centered around the P(t)-adic LC (with P(t) a polynomial). This conjecture reduces to combinatorial properties enjoyed by an infinite array known as a Number Wall. The focus is to develop an ambitious theory to derive P(t)-LC from the patterns emerging from this array.