Generalized Beatty sequences

$A. McD. MERCER Department of Mathematics and Statistics University of Guelph Guelph, Ontario, Canada NIG 2WI (Recieved April 12, 1978) ABSTRACT. A well-known result due to S. Beatty is that if and B are positive irrational numbers satisfying a B- 1 1 then each positive integer is to be found in precisely one of the sequences {[ks]}, {[kB]â} (k-- 1,2,3,...) where [x] denotes the integral part of x. The present note generalizes this result to the case of the pair of sequences {[f(k)]} on the functions f and g. {[g(k)]} with suitable hypotheses The special case f(x) ax, g(x) Bx is the result due to Beatty. KEY WORDS AND PHRASES. Partons Complementary systems of integers, integer sequence. AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES. i. Primary 10A45, Secondary 10L99. The object of this note is to prove the following result. THEOREM. Let f and g be strictly increasing in [0,+ =) with f(O) ;(0) O. Let fâ exist in (0,+ =) and suppose 1 < fâ (x) < 2 there. Suppose also that A. M. MERCER f(k), g(k) (k #(x) + 8(x) 1,2,3,...) are never integral. x in Writing and 8 let [0, + ). Then each positive integer is to be found in pre- cisely one of the sequences {[f(k)]}, {[g(k)]} (k 1,2,3,...) and these sequences consist of distinct integers. This theorem generalizes a well-known result. Namely, if and 8 are positive irrationals satisfying e 8- i 1 then each positive integer is to be found in precisely one of the sequences {[ks]}, {[kS]} (k 1,2,3,...). [I] and has lead [2] who also gives This latter result appears to be due originally to Beatty to notes by several authors, one of the latest being Fraenkel a fairly complete bibliography. Two additional references are given at the end of the present note. In [2] Fraenkel gives also a simple proof of a result due to Lambek and Moser. This result characterizes "two-way splittings" of the set of integers in the sense that it indicates how the second sequence can be generated when an origional sequence of integers is given. nature from the one we prove here. Their result is of quite a different Our present theorem seems to represent a generalization previously overlooked. In this section we prove the theorem stated in Section 1. The proof falls into three parts. PROOF. (a) The two sequences of the theorem are disjoint. Suppose on the contrary that there are integers m and n such that [f(m)] [g(n)]. g(n) Hence Then for some e, 8 (0,I) and M + 8. (M + e) + 8 (M + 8). Let us assume, without loss of generality, that e increasing functions we get (M +) + O(M+) _ _ m+n some integer M we will have f(m) M + e, and 8 are strictly (M+) + O(M+) That is M + e <_- _-< M + 6 ,(0 < e < < i) which is impossible (b) No integer is absent from both sequences. Suppose on. the contrary that the integer N is omitted from both sequences. Let k be any integer and consider [f(k)] and [f(k+l)]. If If(k)] Q then f(k) Hence f(k+l) Q + e (0 < e < I) f(k) + f(k+l) Q + e f(k) + fâ(k+y) (Say) (0<< I) Q + A where 1 < A < 3, 1 < fâ(x) < 2 by hypothesis. Hence [f(k+l)] or Q + 2. It follows that if an integer N is omitted from both sequences then it must lle between the integers If(n)] and [f(n+l)] for some n. Now let m be the greatest integer such that and N [g(m)] [f(n)]. N [f(n)] + I [f(n+l)] we must have [g(m)] <- N f(n) < N 2 and [g(m+l)] __> + 2. (I) That is and g(m) < N- I g(m+l) > N + 2 f(n+l) > N + I (2) From (I) we get Therefore + + n < (N) n +{8(N) + 0(N) 8(N)} <N is strictly increasing this gives m + 1 _-< N (N+I) + 8(N+2) (3) From (2) we get Therefore + + n n + 2 i > N + {8(N+2) 8(N+I)} (4) is strictly increasing this gives m n > N The inequalities (3) and (4) provide the desired contradiction and the proof of part (b) is complete. (c) Each sequence consists of distinct elements. In the proof of part (b) we have seen that because fâ (x) < 2 then [f(k)] [f(k+l)]. Now I < fâ(x) < 2 implies 2 < gâ(x) < and this, in a A. M. MERCER similar way, implies that [g(k)] [g(k+l)]. This completes the proof of (c) and hence of the theorem. We conclude this note by mentioning that the special case: f(x) sx, g(x) 8x, s irrational, e 8- i I, 1 < e < 2, is the classical result due to Beatty.$

Generalized Beatty sequences

Hindawi Publishing Corporation – Aug 23, 2007