2025/02/15 by Yuto Nakajima, Hiroki Takahasi, Nakajima, Yuto +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2502.10902
openalex publication_date 2025/02/15 · openalex created_date 2025/02/19 · openalex updated_date 2026/07/28
We build a bridge from density combinatorics to dimension theory of continued fractions. We establish a fractal transference principle that transfers common properties of subsets of \mathbb N with positive upper density to properties of subsets of irrationals in (0,1) for which the set \an(x)\colon n∈\mathbb N\ of partial quotients induces an injection n∈\mathbb N↦ an(x)∈\mathbb N. Let (*) be a certain property that holds for any subset of \mathbb N with positive upper density. The principle asserts that for any subset S of \mathbb N with positive upper density, there exists a set ES of Hausdorff dimension 1/2 such that the set \bigcupn∈\mathbb N\bigcapx∈ ES\an(x)\∩ S has the same upper density as that of S, and thus inherits property (*). Examples of (*) include the existence of arithmetic progressions of arbitrary lengths and the existence of arbitrary polynomial progressions, known as Szemerédi's and Bergelson-Leibman's theorems respectively. In the same spirit, we establish a relativized version of the principle applicable to the primes, to the primes of the form y2+z2+1, to the sets given by the Piatetski-Shapiro sequences.