A new lower bound for \(\|(3/2)^k\|\). (English) Zbl 1127.11049

In 1957, Mahler proved that the inequality \(\| (p/q)^k\| <C^k,\) where \(p>q>1\) are relatively prime integers, has only finitely many solutions in positive integers \(k\) for any \(C\) satisfying \(0<C<1.\) His proof is ineffective and gives no bound for \(k.\) For applications to Waring’s problem, the lower bound \(\| (3/2)^k\| >0.75^k\) for \(k \geq 5\) would give an exact expression for the least integer \(g(k)\) such that every positive integer can be expressed by a sum of at most \(k\) positive powers. Thanks to some earlier papers of Baker and Coates, Beukers, the reviewer and, finally, Habsieger, the best known effective estimate up to now was \(\| (3/2)^k\| >0.577^k\) for each \(k \geq K_0.\)
In this paper, the author improves this inequality to \(\| (3/2)^k\| >0.5803^k\) for each \(k \geq K,\) where \(K\) is an effective constant. Although the improvement is numerically quite small, the proof is based on constructing a somewhat different sequence of approximations. It involves some very precise computations and, in addition, a careful selection of parameters. In conclusion, using the same method the author presents several sequences of approximations implying the estimates \(\| (4/3)^k\| > 0.4914^k\) for \(k \geq K_1\) and \(\| (5/4)^k\| > 0.5152^k\) for \(k \geq K_2\) improving (at least asymptotically) upon previous results of Bennett.


11J71 Distribution modulo one
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