Fractional parts of powers of rational numbers

Abstract

AbstractThe author uses Padé approximation techniques and an elementary lemma on primes dividing binomial coefficients to sharpen a theorem of F. Beukers on fractional parts of powers of rationals. In particular, it is proven that ‖((N+ l)/N)k‖ > 3–k holds for all positive integers N and k satisfying 4 ≤ N ≤ k · 3k. Other results are described including an effective version of a theorem of K. Mahler for a restricted class of rationals.