Resumen de Binary sequences generated by sequences \n\alpha\, n=1,2,\dots
Stefan Porubsky, Oto Strauch
Let ® be an irrational number, I be a subinterval of the unit interval (0; 1), and fxg denote the fractional part of x. In this paper we shall study arithmetical properties of the set A = fn 2 N; fn®g 2 Ig and pseudorandom character of the sequence xn, n = 1; 2; : : : , where xn = 1 when fn®g 2 I, and xn = ¡1 otherwise. We prove, among others, that the gaps between successive elements of A are at most of three lengths, a, b and a + b also in the case of an arbitrary interval I ½ (0; 1), thereby extending the known Slater's results for intervals of the type I = (0; t) with t < 1=2.
Further we exactly describe the set of positive integers which are not equal to a di®erence of two arbitrary elements from A and we prove that A contains in¯nite double-arithmetic progressions. Then we ¯nd a new lower estimate of the Mauduit{ S¶arkÄozy well distribution meaasure of xn for an arbitrary interval I. We also prove that the sequence xn is Sturmian for every interval I of length f®g or 1 ¡ f®g in the sense that the number of 1's in any pair of ¯nite subsegments of the same length occurring in xn can di®er by at most one. We prove (Theorem ??) that if jIj · 1=2 then any subsequence of xn of the form xn+kK, k = 1; 2; : : : , splits into consecutive blocks of 1's and blocks of ¡1's whose lengths also di®er by at most one. The proofs employ two geometric ideas: (i) a transposition of subintervals (cf. Lemma ??) of I to construct arithmetic progressions of the set A, (ii) properties (cf. Lemma ??) of line segments of the intersection of the graph of the sawtooth function x+fk®g with I £I to answer the question when two elements fn®g and f(n+k)®g simultaneously fall into I. This technique gives, for instance, a new proof of the mentioned Slater's three gap theorems.
