Open AccessL q -spectra of self-affine measures: closed forms, counterexamples, and split binomial sums
Author(s) -
Jonathan M. Fraser,
Lawrence D. Lee,
Ian D. Morris,
Han Yu
Publication year - 2021
Publication title -
nonlinearity
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.571
H-Index - 90
eISSN - 1361-6544
pISSN - 0951-7715
DOI - 10.1088/1361-6544/ac14a2
Subject(s) - mathematics , binomial (polynomial) , counterexample , diagonal , affine transformation , conjecture , combinatorics , binomial theorem , exponential function , expression (computer science) , spectral line , spectrum (functional analysis) , planar , pure mathematics , discrete mathematics , mathematical analysis , statistics , geometry , physics , computer graphics (images) , astronomy , quantum mechanics , computer science , programming language
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 non-trivial closed form bounds for the L q -spectra, which in certain cases yield sharp results.