2025/12/08 by Li, Wenxia, Wang, Zhiqiang, Zhao, Jiuzhou
Mathematics · #Rings, Modules, and Algebras #Mathematical Dynamics and Fractals #Holomorphic and Operator Theory
paper · doi:10.48550/arxiv.2512.07139
Let K be an imaginary quadratic field and let OK be the ring of algebraic integers of K. For α∈ OK with |α| > 1, define Dα= \bigcupn=0^∞ (OK)/(αn). For β∈ OK with |β|>1 and a finite subset A ⊂ OK, define Sβ,A = \ ∑k=1∞ (ak)/(βk): ak ∈ A ∀ k ∈ ℕ \. Suppose that α and β are relatively prime. In this paper, we show that if dimH Sβ,A < 1, then the intersection Dα∩ Sβ,A is a finite set. In general, the threshold for the Hausdorff dimension of Sβ,A is sharp. If we further assume that OK is a unique factorization domain and that α and α are relatively prime, then we establish the finiteness of the intersection under the weaker condition dimH Sβ,A < 2. This extends the previously known results on the real line.