New bounds on the length of finite pierce and Engel series

Erdös, P.; Shallit, J. O.

Séminaire de théorie des nombres de Bordeaux, Série 2, Tome 3 (1991) no. 1, pp. 43-53.

Every real number $x,0<x\le 1$, has an essentially unique expansion as a Pierce series :

$$x=\frac{1}{x^1}-\frac{1}{x^1{x}^2}+\frac{1}{x^1{x}^2{x}^3}-\cdots$$

where the $x_i$ form a strictly increasing sequence of positive integers. The expansion terminates if and only if $x$ is rational. Similarly, every positive real number $y$ has a unique expansion as an Engel series :

$$y=\frac{1}{y^1}-\frac{1}{y^1{y}^2}+\frac{1}{y^1{y}^2{y}^3}+\cdots$$

where the $y_i$ form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi be not eventually constant. Again, such an expansion terminates if and only if $y$ is rational. In this paper we obtain some new upper and lower bounds on the lengths of these series on rational inputs $a/b$. In the case of the Engel series, this answers an open question of Erdös, Rényi, and Szüsz. However, our upper and lower bounds are widely separated.

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