Lq-spectra of self-affine measures : closed forms, counterexamples, and split binomial sums
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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.
Fraser , J , Lee , L D , Morris , I D & Yu , H 2021 , ' L q -spectra of self-affine measures : closed forms, counterexamples, and split binomial sums ' , Nonlinearity , vol. 34 , no. 9 , pp. 6331-6357 . https://doi.org/10.1088/1361-6544/ac14a2
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DescriptionFunding: Jonathan Fraser was financially supported by a Leverhulme Trust Research Fellowship (RF-2016-500) and an EPSRC Standard Grant (EP/R015104/1). Lawrence Lee was supported by an EPSRC Doctoral Training Grant (EP/N509759/1). Ian Morris was supported by a Leverhulme Trust Research Project Grant (RPG-2016-194). Han Yu was financially supported by the University of St Andrews.
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