Let ‖x ‖ be the distance from x ∈ R to the nearest integer. The author proves the new lower bound ‖ ( 3 2)k ‖> C k = 0.5803 k for k ≥ K, where K is a certain effective constant. This improves the previous estimate due to L. Habsieger [Acta Arith. 106 (2003), no. 3, 299–309; MR1957111 (2003j:11072)] with C = 0.5770. The proof is based on Padé approximations to a tail of the serie
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