Series of Error Terms for Rational Approximations of Irrational Numbers

Let pn/qn be the n-th convergent of a real irrational number α, and let εn = αqn−pn. In this paper we investigate various sums of the type ∑ m εm, ∑ m |εm|, and ∑ m εmx m. The main subject of the paper is bounds for these sums. In particular, we investigate the behaviour of such sums when α is a quadratic surd. The most significant properties of the error sums depend essentially on Fibonacci numbers or on related numbers. 1 Statement of results for arbitrary irrationals Given a real irrational number α and its regular continued fraction expansion α = 〈 a0; a1, a2, . . . 〉 (a0 ∈ Z , aν ∈ N for ν ≥ 1) , the convergents pn/qn of α form a sequence of best approximating rationals in the following sense: for any rational p/q satisfying 1 ≤ q < qn we have ∣ ∣ ∣ ∣ α− pn qn ∣ ∣ ∣ ∣ < ∣ ∣ ∣ ∣ α− p q ∣ ∣ ∣ ∣ . The convergents pn/qn of α are defined by finite continued fractions pn qn = 〈 a0; a1, . . . , an 〉 .