Fractal transference principle for continued fractions of Laurent series
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914498163507200 |
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| author | Nakajima, Yuto |
| author_facet | Nakajima, Yuto |
| contents | We establish a fractal transference principle for continued fraction expansions over the field of Laurent series. Let $S$ be an infinite subset of the set of all polynomials over a finite field of $q$ elements of positive degree with growth density exponent $α\ge 1$, and let $U \subset S$ be a subset of positive relative upper density. We prove that there exists a subset $E_{S,U}$ of the set of points whose continued fraction digits are pairwise distinct and belong to $S$ such that \[\dim_{\rm H} E_{S,U}=\frac{1}{2α}.\] Moreover, the set of digits appearing in the continued fraction expansions of points in $E_{S,U}$ recovers the relative upper density of $U$ in $S$. We also show that the same construction preserves the relative upper density of the corresponding degree sets in $\mathbb N$. As a consequence, combinatorial statements for subsets of $\mathbb N$ of positive upper density can be transferred to degree sets arising from continued fraction expansions of Laurent series on sets of optimal Hausdorff dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20113 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fractal transference principle for continued fractions of Laurent series Nakajima, Yuto Dynamical Systems Number Theory We establish a fractal transference principle for continued fraction expansions over the field of Laurent series. Let $S$ be an infinite subset of the set of all polynomials over a finite field of $q$ elements of positive degree with growth density exponent $α\ge 1$, and let $U \subset S$ be a subset of positive relative upper density. We prove that there exists a subset $E_{S,U}$ of the set of points whose continued fraction digits are pairwise distinct and belong to $S$ such that \[\dim_{\rm H} E_{S,U}=\frac{1}{2α}.\] Moreover, the set of digits appearing in the continued fraction expansions of points in $E_{S,U}$ recovers the relative upper density of $U$ in $S$. We also show that the same construction preserves the relative upper density of the corresponding degree sets in $\mathbb N$. As a consequence, combinatorial statements for subsets of $\mathbb N$ of positive upper density can be transferred to degree sets arising from continued fraction expansions of Laurent series on sets of optimal Hausdorff dimension. |
| title | Fractal transference principle for continued fractions of Laurent series |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2604.20113 |