A generalization of Beatty's theorem (Q2782155)

Let \(P=\{0=a_0,a_1,a_2,\dots\}\) be a strictly increasing unbounded sequence of real numbers. For real \(x\geq 0\), let \(\lfloor x\rfloor_P\) be the largest member of \(P\) that does not exceed \(x\), and for \(x>0\) define \(\rfloor x\lfloor_P\) as \(\lfloor x\rfloor_P\) if \(x\not\in P\) and as \(a_{i-1}\) provided \(\lfloor x \rfloor_P=a_i\). Finally, let \(N_t=\{0,t,2t,3t,\dots\}\) for \(t>0\) and \(N_t^+=N_t\setminus\{0\}\). The main result of the paper extends the known Beatty's theorem [Problem 3173, Am. Math. Mon. 33, 159 (1926); ibid. 34, 159 (1927)] on complementary sequences:NEWLINENEWLINENEWLINELet \(F\) and \(G\) be real, continuous, strictly increasing functions with domains \([0,\infty)\) satisfying \(F(0)=G(0)=0\) and \(\lim_{x\to\infty}(F(x)+G(x))=\infty\). For all \(t>0\), let \(P_t=\{0,(F+G)^{-1}(t), (F+G)^{-1}(2t),(F+G)^{-1}(3t),\dots\}\), and \(P^+_t=P_t\setminus\{0\}\). Then the two sequences \(A_t\) and \(B_t\) defined by \(A_t=\{\rfloor(F^{-1})(n)\lfloor_{P_t}:n\in N_t^+\}\) and \(B_t=\{\lfloor(G^{-1})(n)\rfloor_{P_t}:n\in N_t^+\}\) partition \(P^+_t\) for all \(t>0\). Also the elements of \(A_t\) are distinct and the elements of \(B_t\) are distinct.NEWLINENEWLINENEWLINEThis is not only a considerable extension of Beatty's original discovery, but also of its generalization by \textit{J. Lambek} and \textit{L. Moser} [Am. Math. Mon. 61, 454-458 (1954; Zbl 0056.26904)] in which case one of the functions \(F\) and \(G\) determines the other. Then the authors prove specializations of this result for the so-called complementary functions \(f\) and \(g\), which are strictly increasing real functions defined on \([0,\infty)\) with \(f(0)=g(0)=0\) and \(f(x)+g(x)=x\) for all \(x\geq 0\). In this case the set to be partitioned is \(N_t^+\).