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

We study L^q-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 L^q-spectrum. As a further application we provide examples of self-affine measures whose L^q-spectra exhibit new types of phase transitions. Finally, we provide new nontrivial closed form bounds for the L^q-spectra, which in certain cases yield sharp results.