Badly approximable points on self-affine sponges and the lower Assouad dimension

We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. For self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both the case of Sierpiński sponges/carpets (also known as Bedford–McMullen sponges/carpets) and the case of Barański carpets. We use the fact that the lower Assouad dimension of a hyperplane diffuse set constitutes a lower bound for the Hausdorff dimension of the set of badly approximable points in that set.

SYMBOLIC AND GEOMETRIC LOCAL DIMENSIONS OF SELF-AFFINE MULTIFRACTAL SIERPINSKI SPONGES IN ℝd

In this paper we study the multifractal structure of a certain class of self-affine measures known as self-affine multifractal Sierpinski sponges. Multifractal analysis studies the local scaling behaviour of measures. In particular, multifractal analysis studies the so-called local dimension and the multifractal spectrum of measures. The multifractal structure of self-similar measures satisfying the Open Set Condition is by now well understood. However, the multifractal structure of self-affine multifractal Sierpinski sponges is significantly less well understood. The local dimensions and the multifractal spectrum of self-affine multifractal Sierpinski sponges are only known provided a very restrictive separation condition, known as the Very Strong Separation Condition (VSSC), is satisfied. In this paper we investigate the multifractal structure of general self-affine multifractal Sierpinski sponges without assuming any additional conditions (and, in particular, without assuming the VSSC).