Dichotomy Law for a Modified Shrinking Target Problem in Beta Dynamical System

Let φ:N→(0,1] be a positive function. We consider the size of the set Ef(φ):={β>1:|Tβn(x)−f(β)|<φ(n)i.o.n}, where “i.o.n” stands for “infinitely often”, and f:(1,∞)→[0,1] is a Lipschitz function. For any x∈(0,1], it is proved that the Hausdorff measure of Ef(φ) fulfill a dichotomy law according to lim supn→∞logφ(n)n=−∞ or not, where Tβ is the β-transformation. In ergodic theory, the phenomenon of shrinking targets is crucial for understanding the long-term behavior of systems. By studying the shrinking target problem of the β dynamical system, we can reveal the relationship between randomness and determinism, which is significant for constructing more complex mathematical models. Moreover, there is a close connection between the β transformation and number theory. Investigating the contraction target problem helps uncover new properties and patterns in number theory, advancing the development of this field. In this work, we establish a significant relationship between the decay rate of the positive function φ(n) and the structural properties of the set Ef(φ). Specifically, we show that: The Hausdorff dimension of Ef(φ) either vanishes or is positive based on the behavior of φ(n) as n approaches infinity. The establishment of this dichotomy can help us more effectively understand the geometric characteristics and dynamical behavior of the system, thereby aiding our acceptance and comprehension of complex theories. Researching this shrinking target problem can help us uncover new properties in number theory, leading to a better understanding of the structure of numbers and promoting the development of related fields in number theory.