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SIMULTANEOUS DIOPHANTINE APPROXIMATION USING PRIMES A. BALOG AND J. FRIEDLANDER 1. Introduction It was proved by Vinogradov [5] that there exists a positive 9 such that for every real irrational a the inequality \\ap\\ < p~ has infinitely many solutions for primes p. (Here, as usual, ||/|| denotes the distance from t to the nearest integer.) The quantitative value of 9 has subsequently been sharpened by Vaughan [4] and to 9 = 3/10 by Harman [2]. See also Balog [1]. Here we shall investigate the case of simultaneous approximations, namely the question of whether, for given a , ...,a , and 9, there exist infinitely many primes p x fc such that the inequalities \\oc p\\ <p~ , (i = \,2,...,k) all hold. We consider k- tuples of reals (a ..., a ) which satisfy the compatibility condition: 19 fc k k If/»,eZ( l </<ik)an d £ A,a,eQ then £ A,a,eZ. (*) It is obvious that this condition of compatibility is necessary in order for there to be good approximations. Indeed, if r and s are co-prime integers with s > 1 and if r 5 w tn max Yjt-i^i^i — / ' * l^ l = h then
Bulletin of the London Mathematical Society – Wiley
Published: Jul 1, 1988
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