On the vector space of the automatic reals

AbstractA sequence (an)n ⩾ 0 is said to be k-automatic if an is a finite-state function of the base-k digits of n. We say a real number is (k, b)-automatic if its fractional part has a base-b expansion that forms a k-automatic sequence, and we denote the set of all such numbers as L(k,b). Lehr (Theoret. Comput. Sci. 108 (1993) 385–391) proved that L(k, b) forms a vector space over Q. In this paper we give a shortened version of the proof of Lehr's result and, answering a question of Bach, show that the dimension of the vector space L(k, b) is infinite.We also give an example of a transcendental number such that all of its positive powers are automatic. The proof requires examining the coefficient of Xn in the formal power series (X + X2 + X4 + X8 + …)r. Along the way we are led to examine several sequences of independent combinatorial interest.Finally, solving an open problem, we show that the automatic reals are not closed under (1) product; (2) squaring; and (3) reciprocal

Compositions into Powers of bb: Asymptotic Enumeration and Parameters

For a fixed integer base $b\geq2$, we consider the number of compositions of $1$ into a given number of powers of $b$ and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of $b$. We study the asymptotic growth of those numbers and give precise asymptotic formulae for them, thereby improving on earlier results of Molteni. Our approach uses generating functions, which we obtain from infinite transfer matrices. With the same techniques the distribution of the largest denominator and the number of distinct parts are investigated

Words and Transcendence

Is it possible to distinguish algebraic from transcendental real numbers by considering the $b$-ary expansion in some base $b\ge2$? In 1950, \'E. Borel suggested that the answer is no and that for any real irrational algebraic number $x$ and for any base $g\ge2$, the $g$-ary expansion of $x$ should satisfy some of the laws that are shared by almost all numbers. There is no explicitly known example of a triple $(g,a,x)$, where $g\ge3$ is an integer, $a$ a digit in $\{0,...,g-1\}$ and $x$ a real irrational algebraic number, for which one can claim that the digit $a$ occurs infinitely often in the $g$-ary expansion of $x$. However, some progress has been made recently, thanks mainly to clever use of Schmidt's subspace theorem. We review some of these results

Representation of a 2-power as sum of k 2-powers: the asymptotic behavior

A $k$-representation of an integer \l is a representation of \l as sum of $k$ powers of $2$, where representations differing by the order are considered as distinct. Let \W(\sigma,k) be the maximum number of such representations for integers \l whose binary representation has exactly $\sigma$ non-zero digits. \W(\sigma,k) can be recovered from \W(1,k) via an explicit formula, thus in some sense \W(1,k) is the fundamental object. In this paper we prove that (\W(1,k)/k!)^{1/k} tends to a computable limit as $k$ diverges. This result improves previous bounds which were obtained with purely combinatorial tools