In this note the distribution of the approximation coeecients n, associated with the regular continued fraction expansion of numbers x 2 0; 1), is given under extra conditions on the numerators and denominators of the convergents pn=qn. Similar results are also obtained for S-expansions. Further, a Gauss-Kusmin type theorem is derived for the regular continued fraction expansion under these ext...

The nearest integer continued fractions of Hurwitz, Minnegerode (NICF-H) and in Perron’s book Die Lehre von den Kettenbrüchen (NICF-P) are closely related. Midpoint criteria for solving Pell’s equation x2 − Dy2 = ±1 in terms of the NICF-H expansion of √ D were derived by H. C. Williams using singular continued fractions. We derive these criteria without the use of singular continued fractions. ...

This paper investigates the quadratic irrationals that arise as periodic points of Gauss type shift associated to odd continued fraction expansion. It is shown these numbers, which we call O-reduced, when ordered by length closed primitive geodesic on some modular surface Γ∖H, are equidistributed with respect Lebesgue absolutely continuous invariant probability measure Odd shift.

Abstract. We investigate a collection of orthonormal functions that encodes information about the continued fraction expansion of real numbers. When suitably ordered these functions form a complete system of martingale differences and are a special case of a class of martingale differences considered by R. F. Gundy. By applying known results for martingales we obtain corresponding metric theore...

An old problem adressed by Khintchin [15] deals with the behaviour of the continued fraction expansion of algebraic real numbers of degree at least three. In particular, it is asked whether such numbers have or not arbitrarily large partial quotients in their continued fraction expansion. Although almost nothing has been proved yet in this direction, some more general speculations are due to La...

Let a, b, c, d be complex numbers with d 6= 0 and |q| < 1. Define H1(a, b, c, d, q) := 1 1 + −abq + c (a + b)q + d + · · · + −abq + cq (a + b)qn+1 + d + · · · . We show that H1(a, b, c, d, q) converges and 1 H1(a, b, c, d, q) − 1 = c − abq d + aq P∞ j=0 (b/d)(−c/bd)j q (q)j(−aq/d)j P∞ j=0 (b/d)(−c/bd)j q (q)j(−aq/d)j . We then use this result to deduce various corollaries, including the followi...

The continued fraction expansion of a real number is a fundamental and revealing expansion through its connection with Euclidean algorithm and with ``best'' rational approximations (see [HW]). At the same time, it is very poorly understood for some interesting numbers. We know that it is essentially unique and finite (i.e., terminating) exactly for rational numbers and periodic exactly for quad...

There is a class of quadratic number fields for which it is possible to find an explicit continued fraction expansion of a generator and hence an explicit formula for the fundamental unit. One therewith displays a family of quadratic fields with relatively large regulator. The formula for the fundamental unit seems far simpler than the continued fraction expansion, yet the expansion seems neces...