Number Systems, Spectra and Rational Fractal Tiles

The present application is devoted to the research on several interconnected topics that belong to the branch of pure mathematics called numeration. Numeration is an amalgamation of number theory (number systems in particular), discrete dynamical systems and the study of fractal sets (that, in turn, invokes methods from geometry, measure theory and topology). In the most general sense, it is a science that studies the finite and infinite representations of numbers (integer, real or complex) which can be constructed from a finite digit set by using a fixed power basis. The questions what digit sets and what bases are needed to represent a given number in the optimal way lead naturally to almost linear dynamical mappings and their fractal attractors. In this project, we focus on the following topics: A-adic rational fractal tiles; Lattice number systems with non-contractive matrix basis; Multiple-Shift Radix Systems (MSRS); Attractors of symmetric SRS related to number systems with symmetric digit sets; Tiling digit sets for number systems small positive integer basis; 0-representations in the spectra of algebraic numbers with unimodular conjugates. The project will be led jointly by the applicant and co-applicant at Montanuniversität Leoben in collaboration with several well established experts from abroad. Mathematical tools like the harmonic analysis on projective limits of locally compact groups for the transfer operators or the study of the adjacency matrices of finite automata related to spectra of numbers, aided by computational experiments with computer algebra systems. Expected results for each of the listed scientific topics are: Comprehensive general theory of A-adic rational tiles; Existence and characterization the lattice number system with non-contractive basis matrix; Periodic analogs of the discretized rotations; Characterization of periodic integral points in symmetrical number systems of low dimensions; Formulas for the tiling digit sets for bases of small size & small number of prime factors; The spectral radius for the adjacency matrix of the 0-recognition automaton in the spectra with non-Pisot bases, the number-of-multiples for the polynomials with restricted coefficients and the finiteness of the spectra intersection set. The problems are interconnected in the sense that any advancement in one topic could lead to improvements in other directions. The solution to each problem considered in this project (if achieved) would be a significant contribution to the state of the art in the field.

The numbers are omnipresent in our everyday lives. We use numbers, written in traditional base-10 numeration systems like 4593 = 4103+5102+910+3. Our computers operate in binary system: the same number 4593 reads as 1000111110001b. Mathematicians also consider more sophisticated numeration systems, where the decimal base a=10 or a binary base a=2 is replaced by an algebraic number, like 3+2, or the number systems whose digits are vectors, like (1, -1), and the base is no longer a number, but a matrix, say, [1, -2] [3, 4]. While at the first sight such number systems seem to be very exotic, they might have very important applications in real life, for instance, to encode the online data streams, in 2D and 3D image data file compression. Together with such complicated number systems comes the question what quantities can or cannot be represented in these number systems, how many different representations exist, and how these representations can be visualized. The present FWF Lise Meitner project M2259 Number systems, spectra and rational fractal tiles deals with exactly these questions, and the problems related to this field. While relatively simple to ask, these questions quickly lead to very sophisticated areas of mathematics, like p-adic numbers, fractal geometry or abstract harmonic analysis on localy compact groups. The main results of the present project include the establishment of the theoretical framework of the digit systems with rational matrix bases that satisfy the so called finiteness property, the multi-tiling property associated to expanding rational matrices and standard digit sets. Related questions, like the patterns in {-1, 1} and {0, 1} polynomials with specified zero distributions, or the properties of Pisot and Salem numbers, which are very important in number systems, were also investigated. The research on this project resulted in 3 published papers and 1 preprint submitted for publication in peer-reviewed mathematical journals, with 3 more manuscripts in different degrees of completeness still in preparation. Accompanying the papers, 2 open source libraries of software designed to carry out the mathematical computations on the subjects of these papers have been released.