For x ∈ (0, 1], let x = [d1(x),d2(x),…,dn(x),…]L be its Luroth expansion, and let {pn(x)/qn(x)}n≥1 be the sequence of convergents of x. Define the exceptional sets E(β) = x ∈ (0, 1]: limn→∞log dn+1...

Exact Dimensions of Exceptional Sets in Luroth Expansions

We introduce a family of random $c$-Luroth transformations $\{L_c\}_{c \in [0, \frac12]}$, obtained by randomly combining the standard and alternating Luroth maps with probabilities $p$ and $1-p$, $0 0$, we compute the density of the stationary measure and we use it to study the typical speed of convergence of the approximants and the digit frequencies.

On approximation by random L\"uroth expansions.

For x ∈ [0, 1), let x = O1(q 1(x),q2(x),…,qk(x),…) be its first Ostrogradsky series with digits {qn(x)}n≥1, {Pn(x) Qn(x)}n≥1 be its O1-convergents. We calculate the Hausdorff dimensions of the sets...

Hausdorff dimensions of some exceptional sets in the first ostrogradsky series

For any x ∈ (0, 1], let x = 1 d1 + 1 d1(d1 − 1)d2 + ⋯ + 1 d1(d1 − 1)⋯dn−1(dn−1 − 1)dn + ⋯ be its Luroth expansion with digits {dj ≥ 2,j ≥ 1}. Let ψ : ℕ → ℝ+ be a function satisfying ψ(n)/n →∞ as n ...

Luming Shen

Papers

Hausdorff dimension of the set concerning with borel-bernstein theory in lüroth expansions