Statistical Properties of digital expansions

Digital expansions of natural numbers are used in many different areas of mathematics (to say nothing of the binary representation of numbers in every computer). Especially they occur in various algorithms, e.g. for computing powers with large exponents, and for generating random numbers. Therefore there is a demand for investigating statistical properties of digital expansions. This research project aims at applications as well as at theoretical aspects, most of which are of a number-theorical nature.

Most of our methods come from combinatorics, analytic number theory, complex analysis, and probability theory. Digital expansions based on linear recursions surprisingly raise geometric questions concerning dynamical processes generated by torus automorphisms.

Selected Publications:

  • M. Drmota and J. Gajdosik. The distribution of the sum-of-digits function. J. Théor. Nombres Bordeaux 10/1 (1998), 17-32. [MR].
  • M. Drmota and J. Gajdosik. The parity of the sum-of-digits-function of generalized Zeckendorf representations. Fibonacci Quart. 36/1 (1998), 3-19. [MR].
  • M. Drmota and M. Skalba. Rarified sums of the Thue-Morse sequence. Trans. Amer. Math. Soc. 352/2 (2000), 609-642. [MR].
  • M. Drmota and M. Skalba. The parity of the Zeckendorf sum-of-digits function. Manuscripta Math. 101/3 (2000), 361-383. [MR].
  • M. Drmota. The distribution of patterns in digital expansions. In F. Halter-Koch and R. F. Tichy, editors, Algebraic number theory and Diophantine analysis (Graz, 1998), pages 103-121. de Gruyter, Berlin, 2000. [MR].
  • M. Drmota and G. Larcher. The sum-of-digits-function and uniform distribution modulo 1. J. Number Theory 89/1 (2001), 65-96. [MR].
  • M. Drmota. The joint distribution of q-additive functions. Acta Arith. 100/1 (2001), 17-39. [MR].
  • W. Steiner. Parry expansions of polynomial sequences. Integers 2 (2002), Paper A14, 28 pp. (electronic). [MR], [pdf].
  • M. Fuchs. Digital expansion of exponential sequences. J. Théor. Nombres Bordeaux 14/2 (2002), 477-487. [MR], [pdf].
  • M. Drmota and W. Steiner. The Zeckendorf expansion of polynomial sequences. J. Théor. Nombres Bordeaux 14/2 (2002), 439-475. [MR], [pdf].
  • M. Fuchs. On metric Diophantine approximation in the field of formal Laurent series. Finite Fields Appl. 8/3 (2002), 343-368. [MR], [pdf].
  • M. Drmota, M. Fuchs, and E. Manstavičius. Functional limit theorems for digital expansions. Acta Math. Hungar. 98/3 (2003), 175-201. [MR], [pdf].
  • M. Fuchs. On a problem of W. J. LeVeque concerning metric Diophantine approximation. Trans. Amer. Math. Soc. 355/5 (2003), 1787-1801 (electronic). [MR], [pdf].
  • W. Steiner. Generalized de Bruijn digraphs and the distribution of patterns in alpha-expansions. Discrete Math. 263/1-3 (2003), 247-268. [MR], [pdf].