Metric Properties and Exceptional Sets of the Oppenheim Expansions over the Field of Laurent Series

Metric properties for p-adic Oppenheim series expansions

“…The corresponding results for Oppenheim series expansions of Laurent series have been obtained by Fan and the author [3].…”

“…Then define A 1 = A − a 0 . As in [9], [10], if A n = 0 with v(A n ) ≥ 1 (n ≥ 1) is already defined, then define the "digit" a n = 1/A n and put (3) A n+1 = A n − 1 a n s n (a n ) r n (a n ) .…”

Approximation orders of real numbers by $$\beta $$-expansions

“…The problem on the approximation orders is a longstanding topic in mathematics, for example, the approximation of functions or the numbers. The approximation problems on the representations of real numbers have been widely investigated, see [11,21,33] for continued fractions, see [13,14] for Oppenheim expansions, see [3,9] for Lüroth expansions.…”

On singularity of distribution of random variables with independent symbols of Oppenheim expansions

“…This paper is devoted to the development of probabilistic theory of Oppenheim expansions of real numbers which contains many important expansions as rather special cases. Let us mention that many authors studied normal properties of real numbers in terms of digits of their Oppenheim expansion and the Hausdorff dimension of corresponding exceptional sets (see, e.g., [13,15,44,45,24]). In Section 2 we develop approach which has been invented by G. Torbin to study normal properties of the Ostrogradsky-Sierpinski-Pierce expansion [40] and get general results on normal properties of Oppenheim expansions.…”