Lq-spectra of self-affine measures: closed forms, counterexamples, and split binomial sums

Jonathan Fraser, Lawrence D. Lee, Ian D. Morris, Han Yu

Research output: Contribution to journalArticlepeer-review


We study Lq-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 Lq-spectrum. As a further application we provide examples of self-affine measures whose Lq-spectra exhibit new types of phase transitions. Finally, we provide new nontrivial closed form bounds for the Lq-spectra, which in certain cases yield sharp results.
Original languageEnglish
Pages (from-to)6331-6357
Number of pages27
Issue number9
Early online date2 Aug 2021
Publication statusPublished - Sept 2021


  • Fractals
  • Lq-spectra
  • Self-affine measures


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