On the Approximation of Irrational Numbers by Rational Numbers

Unsymmetrical Approximation of Irrational Numbers

1 A 1 < £ < , 5B B SB so that we have the classical theorem of Hurwitz. For other values of r, approximations from both sides are permitted, but the errors allowed on the two sides are different; hence the term unsymmetrical approximation. The result here was new, and is so related to Hurwitz's inequality that one side is strengthened and the other weakened. Notice that the result for r > l is ...

On Irrational and Transcendental Numbers

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 nu...

On the Approximation to Algebraic Numbers by Algebraic Numbers

Let n be a positive integer. Let ξ be an algebraic real number of degree greater than n. It follows from a deep result of W. M. Schmidt that, for every positive real number ε, there are infinitely many algebraic numbers α of degree at most n such that |ξ−α| < H(α)−n−1+ε, where H(α) denotes the näıve height of α. We sharpen this result by replacing ε by a function H 7→ ε(H) that tends to zero wh...

On the Approximation of Irrational Numbers with Rationals Restricted by Congruence Relations

Usually this theorem is proved by using continued fractions; see Theorem 193 in [2]. S. Hartman [3] has restricted the approximating numbers f to those fractions, where u and v belong to fixed residue classes a and b with respect to some modulus s. He proved the following: For any irrational number £, any s > 1, and integers a and b, there are infinitely many integers u and v > 0 satisfying 2s (1)