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Extra resources for Collected papers of G.H. Hardy; including joint papers with J.E. Littlewood and others

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Important for us to cotisider very carefully the ideas involved in them with a view to seletit’ing those which lend themselves most readily to generalisation. For example, it is essential that our proof should make no appeal to the theory of continued fractions. (a). The first proof is due to. KRONECKER. It follows 1 . 10, with m = I, or from the theory of continued fractions, an arbitrarily large q such that (I. III) q 0 - p = d/q. to express It is possible nearest to q a, in the form where from the result of that we can find rt and n, are integers, any integer, and 1nl (q/z.

It is (as is well known) possible to choose 0 so that (n! e)--o. unique limit. At the end of the paper we shall return to the general problem. The immediate object with which this paper was begun, however, was to determine whe- 29 157 Some problems of Diopbantine Approximation. ther the relation (I 002) always holds (if 0 is irrational) when f (n) is a po’wer of n, and we shall be for the most part concerned with this special form of f(n). I . 01. 01. If 0, Fo,, -- . Ornare linearly independent irrationals (i.

The first of these (section x . I) consists of a discussion and. proof of KRONECKER’S theorem. We have thought it worth while to devote some space to this for two reasoni. 011 proceeds by induction from k to k + I, and it seems desirable for the s’ake of completeness *to give some account of the methods by which the theorem is established in the case k= 1. In the second placemthe theorem for this case possessesan interest and importance rsufficient to justify any attempt to throw new light upon it; and the ideas involved in the various proofs which we shall discuss are such as are important in the further developmentIs of the theory.

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