Finite-State Dimension and Real Arithmetic

We use entropy rates and Schur concavity to prove that, for every integer k >= 2, every nonzero rational number q, and every real number alpha, the base-k expansions of alpha, q+alpha, and q*alpha all have the same finite-state dimension and the same finite-state strong dimension. This extends, and gives a new proof of, Wall's 1949 theorem stating that the sum or product of a nonzero rational number and a Borel normal number is always Borel normal.Comment: 15 page

Equidistribution from Fractals

We give a fractal-geometric condition for a measure on [0,1] to be supported on points x that are normal in base n, i.e. such that the sequence x,nx,n^2 x,... equidistributes modulo 1. This condition is robust under C^1 coordinate changes, and it applies also when n is a Pisot number and equidistribution is understood with respect to the beta-map and Parry measure. As applications we obtain new results (and strengthen old ones) about the prevalence of normal numbers in fractal sets, and new results on measure rigidity, specifically completing Host's theorem to multiplicatively independent integers and proving a Rudolph-Johnson-type theorem for certain pairs of beta transformations.Comment: 46 pages. v3: minor corrections and elaboration

Unexpected distribution phenomenon resulting from Cantor series expansions

We explore in depth the number theoretic and statistical properties of certain sets of numbers arising from their Cantor series expansions. As a direct consequence of our main theorem we deduce numerous new results as well as strengthen known ones.Comment: 32 page

Visual art inspired by the collective feeding behavior of sand-bubbler crabs

Sand--bubblers are crabs of the genera Dotilla and Scopimera which are known to produce remarkable patterns and structures at tropical beaches. From these pattern-making abilities, we may draw inspiration for digital visual art. A simple mathematical model is proposed and an algorithm is designed that may create such sand-bubbler patterns artificially. In addition, design parameters to modify the patterns are identified and analyzed by computational aesthetic measures. Finally, an extension of the algorithm is discussed that may enable controlling and guiding generative evolution of the art-making process