L^q-spectra of self-affine measures: Closed forms, counterexamples, and split binomial sums

Abstract

<jats:title>Abstract</jats:title> <jats:p>We study <jats:italic>L</jats:italic> <jats:sup> <jats:italic>q</jats:italic> </jats:sup>-spectra of planar self-affine measures generated by diagonal matrices. We introduce a new technique for constructing and understanding examples based on combinatorial estimates for the exponential growth of certain split binomial sums. Using this approach we disprove a theorem of Falconer and Miao from 2007 and a conjecture of Miao from 2008 concerning a closed form expression for the generalised dimensions of generic self-affine measures. We also answer a question of Fraser from 2016 in the negative by proving that a certain natural closed form expression does not generally give the <jats:italic>L</jats:italic> <jats:sup> <jats:italic>q</jats:italic> </jats:sup>-spectrum. As a further application we provide examples of self-affine measures whose <jats:italic>L</jats:italic> <jats:sup> <jats:italic>q</jats:italic> </jats:sup>-spectra exhibit new types of phase transitions. Finally, we provide new non-trivial closed form bounds for the <jats:italic>L</jats:italic> <jats:sup> <jats:italic>q</jats:italic> </jats:sup>-spectra, which in certain cases yield sharp results.</jats:p>