A Greatest Integer Theorem for Fibonacci Spaces

defined by u% + MUQ+J = u^+2The latter is shown to be true in all cases but one, and in slightly revised form in the remaining case. Z A GENERAL ASYMPTOTIC THEOREM With the polynomial fix) = -ag-a-jX an.fx " +x = (x rj) -(x rn), a,integers, r-j real, /y distinct, |/y| < 1 for / > 2, we associate the /7-space C(f) of all (complex) sequences S =is0,sif—J in which SQ, —, sn-i are arbitrary, but having aoSj + '-' + an-fSj+n-i = s1+n; j > 0. The n geometric sequences /?/= \i,n,r?,-\ form a basis for the space C(f), in terms of which an arbitrary integral sequence S may be expressed in the form S = ciRi + + cnRn, i.e., s% = c-frf + +cnr^; fi> 0. Since |/y| < /, / > 2, we may write (1) SQ = cir-j +e%; e% -» 0. These results may be found in [2 ] . That cj (and hence e%) are real is shown in an Appendix. As an immediate consequence, we have the asymptotic Theorem 1. Let F be an arbitrary constant on the open interval (0,1), and S= j sj I an integral sequence of the space C(f). Then for fixed k > 0, one has the greatest integer