Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.06933 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910693571166208 |
|---|---|
| author | Erazo, Harold Lima, Davi Matheus, Carlos Moreira, Carlos Gustavo Vieira, Sandoel |
| author_facet | Erazo, Harold Lima, Davi Matheus, Carlos Moreira, Carlos Gustavo Vieira, Sandoel |
| contents | The Lagrange and Markov spectra $L$ and $M$ describe the best constants of Diophantine approximations for irrational numbers and binary quadratic forms. In 1880, A. Markov showed that the initial portions of these spectra coincide: indeed, $L\cap (0,3) = M\cap (0,3)$ is a discrete set of explicit quadratic irrationals accumulating only at $3$.
In this article, we show that the statement above ceases to be true immediately after $3$: in particular, $L\cap (3,3+\varepsilon)\neq M\cap (3,3+\varepsilon)$ for all $\varepsilon>0$, and thus $\inf(M\setminus L)=3$. In fact, we derive this result as a by-product of lower bounds on the Hausdorff dimension of $(M\setminus L)\cap (3,3+\varepsilon)$ implying that $\liminf\limits_{\varepsilon\to 0} \frac{\dim_H((M\setminus L)\cap(3,3+\varepsilon))}{\dim_H(M\cap (3,3+\varepsilon))}\geq \frac{1}{2}$ and, as it turns out, these bounds are obtained from the study of projections of Cartesian products of almost affine dynamical Cantor sets via an argument of probabilistic flavor based on Baker--Wüstholz theorem on linear forms in logarithms of algebraic numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06933 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | inf(M \ L)=3 Erazo, Harold Lima, Davi Matheus, Carlos Moreira, Carlos Gustavo Vieira, Sandoel Number Theory Dynamical Systems The Lagrange and Markov spectra $L$ and $M$ describe the best constants of Diophantine approximations for irrational numbers and binary quadratic forms. In 1880, A. Markov showed that the initial portions of these spectra coincide: indeed, $L\cap (0,3) = M\cap (0,3)$ is a discrete set of explicit quadratic irrationals accumulating only at $3$. In this article, we show that the statement above ceases to be true immediately after $3$: in particular, $L\cap (3,3+\varepsilon)\neq M\cap (3,3+\varepsilon)$ for all $\varepsilon>0$, and thus $\inf(M\setminus L)=3$. In fact, we derive this result as a by-product of lower bounds on the Hausdorff dimension of $(M\setminus L)\cap (3,3+\varepsilon)$ implying that $\liminf\limits_{\varepsilon\to 0} \frac{\dim_H((M\setminus L)\cap(3,3+\varepsilon))}{\dim_H(M\cap (3,3+\varepsilon))}\geq \frac{1}{2}$ and, as it turns out, these bounds are obtained from the study of projections of Cartesian products of almost affine dynamical Cantor sets via an argument of probabilistic flavor based on Baker--Wüstholz theorem on linear forms in logarithms of algebraic numbers. |
| title | inf(M \ L)=3 |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2411.06933 |