Diophantine approximation with partial sums of power series

Abstract

For each nonzero rational number r, in [1], we considered the problem of approximating G(r) with partial sums of the series (1.1). In the case that an ≡ 1 and s = 1, we asked how well one can approximate e by the partial sums ∑n `=0 1 `! . J. Sondow [6] conjectured that exactly two of these partial sums are also convergents to the continued fraction of e. Among several results, Sondow and K. Schalm [7], proved that for almost all positive integers n, the partial sum ∑n `=0 1 `! is not a convergent to the continued fraction of e. Thus, the probability of obtaining a convergent to the continued fraction of e upon randomly choosing one of the first n partial sums of the power series of e tends to zero as n → ∞. Knowledge of the continued fraction of e, where a is a nonzero integer, and the best possible diophantine approximation of e, discovered by S. Ramanujan [5] and rediscovered by C. S. Davis [4], enabled the authors to prove in [2] that at most Oa(logM) of the first M convergents to the continued fraction of e are also partial sums of the corresponding power series. In [1], we considered general hypergeometric functions pFp(a1, . . . , ap; b1, . . . , bp; r) and showed that among their first N partial sums, no more than O(logN) are convergents to the continued fraction of pFp(a1, . . . , ap; b1, . . . , bp; r). Observe that this result includes e as a special case and that this particular corollary is a dual of the aforementioned result established in [2]. Moreover, when an is a real Dirichlet character or when {an}, n > 0, are the coefficients of an L-series attached to an elliptic curve without complex multiplication, we proved similar theorems in [1]. Lastly, we remark that in [2], we, in fact, proved Sondow’s conjecture. At the focal point of our study in [1] and in this paper are the following two definitions. For any rational number μ = a/b, with (a, b) = 1, consider the height H(μ) of μ, given by H(μ) = max{|a|, |b|}. For any real number α, and any positive real number δ, denote