Sequence Of Denominators Research Articles

Upper Bounds for Double Exponential Sums Along a Subsequence

AbstractWe consider a class of double exponential sums studied in a paper of Sinai and Ulcigrai. They proved a linear bound for these sums along the sequence of denominators in the continued fraction expansion of α, provided α is badly-approximable. We provide a proof of a result, which includes a simple proof of their theorem, and which applies for all irrational α.

Approximation of real numbers with rational number sequences

Let α ∈ R, and let C > max{1, α}. It is shown that if {pn/qn} is a sequence formed out of all rational numbers p/q such that ∣∣∣α− pq ∣∣∣ ≤ 1 Cq2 , where p ∈ Z and q ∈ N are relatively prime numbers, then either {pn/qn} has finitely many elements or lim sup n→∞ log log qn logn ≥ 1, where the points {qn}n∈N are ordered by increasing modulus. This implies that the sequence of denominators {qn}n∈N grows exponentially as a function of n, and so the density of rational numbers which approximate α well in the above sense is relatively low. It has been suggested that one could obtain strong results in number theory by translating the proofs from Nevanlinna theory to number theory using, for instance, Vojta’s dictionary [16]. Applying this method to Nevanlinna’s proof of the second main theorem is said to yield strong results in number theory, including the abc conjecture formulated by Masser and Oesterle. Difficulties arise, however, while trying to find a number-theoretic analogue for the ramification term in the second main theorem, since so far Vojta’s dictionary does not have a translation for the derivative of a meromorphic function; see, e.g., [1, 7, 12]. There is a vast literature on the rational approximations of irrational numbers dating all the way back to Liouville [9], Thue [15], Siegel [13] and others (see [1] for further references). One of the fundamental results in the field is Roth’s theorem, according to which for any algebraic irrational number α there are only finitely many solutions p/q to (1) ∣∣∣α− pq ∣∣∣ ≤ 1 q2+e , where e > 0, and p and q are relatively prime integers [11]. Roth’s theorem is sharp in the sense that the arbitrary e > 0 cannot be deleted in inequality (1). However, Lang [8] has proposed that if one would be able to follow the proof of the second main theorem with a good error term (see, e.g., [6]) to prove Roth’s theorem, then Received by the editors October 22, 2007, and, in revised form, January 9, 2008. 2000 Mathematics Subject Classification. Primary 11J68; Secondary 11J97. The research reported in this paper was supported in part by the Academy of Finland grants #118314 and #210245. c ©2008 American Mathematical Society Reverts to public domain 28 years from publication 107 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Shohat–Favard type theorem for orthogonal series

A function f is considered in the form of its formal series expansion ∑ k = 0 ∞ f k P k where ( P k ) k = 0 ∞ denotes a given system of orthogonal polynomials with respect to a measure on the real line. We prove that the denominators, numerators and residuals of their Frobenius–Padé approximation satisfy a five-term recurrence relation. The converse result is obtained: starting from the sequence of denominators, the function f is reconstructed, i.e. the analogous of the classical Shohat–Favard theorem is done.

Sur le d�veloppement en fractions continues a quotients partiels impairs

Let (B n ) n be the sequence of denominators of convergents, given by the continued fraction expansion with odd partial quotients of an irrational number. For integersm>2 andk, 0≤k<m, we give (almost everywhere) the density of the set of integersn such thatB n is congruent tok modm. This result comes from an ergodic system.

Denominator Sequences of continued fractions I

Let α be an irrational number with simple continued fraction α = (a0, a1, a2,…). The problem studied is that of whether the sequence (qn) of denominators of the convergents pn/qn to α has a subsequence (Bn) = (qin} which is the sequence of denominators of convergents An/Bn to a different number α′. In other words, does there exist a subsequence {qin} which satisfies qio= 1 and For example, the sequence of denominators of convergents to ½(3 − e) is a subsequence of the sequence of denominators of convergents to ε.