## 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.

### MSC:

 11J71 Distribution modulo one
Full Text:

### Online Encyclopedia of Integer Sequences:

Numbers n such that 1.5^n is closer to an integer than 1.5^m for any 0 < m < n.

### References:

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