Combinatorics on Number Walls and the $P(t)$-adic Littlewood Conjecture
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2023
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| author | Robertson, Steven |
| author_facet | Robertson, Steven |
| contents | For any prime $p$ and real number and $α$, the $p$-adic Littlewood Conjecture due to de Mathan and Teulié asserts that \[\inf_{|m|\ge1}|m|_p\cdot |m|\cdot |\left\langleαm\right\rangle|=0.\] Above, $|m|$ is the usual absolute value, $|m|_p$ is the $p$-adic norm and $\left|\left\langle x\right\rangle\right|$ is the distance from $x\in\mathbb{R}$ to the nearest integer. Let $\mathbb{K}$ be a field and $P(t)\in\mathbb{K}[t]$ be an irreducible polynomial. This paper deals with the analogue of this conjecture over the field of formal Laurent series over $\mathbb{K}$, known as the $P(t)$-adic Littlewood Conjecture ($P(t)$-LC). The following results are established:
(1) Any counterexample to $P(t)$-LC for the case $P(t)=t$ generates a counterexample when $P(t)$ is any irreducible polynomial. Since $P(t)$-LC is knwon to be false when $P(t)=t$ and $\mathbb{K}$ has characteristic 0,3,5,7 and 11, one obtains a disproof of the $P(t)$-LC over any such field for any choice of irreducible polynomial $P(t)$.
(2) A Khintchine-type theorem for $t$-adic multiplicative approximation is established, enabling one to determine the measure of the set of counterexamples to $P(t)$-LC with an additional monotonic growth function in the case $P(t)=t$.
(3) The Hausdorff dimension of the same set is shown to be maximal when $P(t)=t$ in the critical case where the growth function is $\log^2$.
These goals are achieved by developing an extensive theory in combinatorics relating $P(t)$-LC to the properties of the so-called number wall of a sequence. This is an infinite array containing the determinant of every finite Toeplitz matrix generated by that sequence. The main novelty of this paper is creating a dictionary allowing one to transfer statements in Diophantine approximation in positive characteristic to combinatorics through the concept of a number wall, and conversely. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_00955 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Combinatorics on Number Walls and the $P(t)$-adic Littlewood Conjecture Robertson, Steven Number Theory For any prime $p$ and real number and $α$, the $p$-adic Littlewood Conjecture due to de Mathan and Teulié asserts that \[\inf_{|m|\ge1}|m|_p\cdot |m|\cdot |\left\langleαm\right\rangle|=0.\] Above, $|m|$ is the usual absolute value, $|m|_p$ is the $p$-adic norm and $\left|\left\langle x\right\rangle\right|$ is the distance from $x\in\mathbb{R}$ to the nearest integer. Let $\mathbb{K}$ be a field and $P(t)\in\mathbb{K}[t]$ be an irreducible polynomial. This paper deals with the analogue of this conjecture over the field of formal Laurent series over $\mathbb{K}$, known as the $P(t)$-adic Littlewood Conjecture ($P(t)$-LC). The following results are established: (1) Any counterexample to $P(t)$-LC for the case $P(t)=t$ generates a counterexample when $P(t)$ is any irreducible polynomial. Since $P(t)$-LC is knwon to be false when $P(t)=t$ and $\mathbb{K}$ has characteristic 0,3,5,7 and 11, one obtains a disproof of the $P(t)$-LC over any such field for any choice of irreducible polynomial $P(t)$. (2) A Khintchine-type theorem for $t$-adic multiplicative approximation is established, enabling one to determine the measure of the set of counterexamples to $P(t)$-LC with an additional monotonic growth function in the case $P(t)=t$. (3) The Hausdorff dimension of the same set is shown to be maximal when $P(t)=t$ in the critical case where the growth function is $\log^2$. These goals are achieved by developing an extensive theory in combinatorics relating $P(t)$-LC to the properties of the so-called number wall of a sequence. This is an infinite array containing the determinant of every finite Toeplitz matrix generated by that sequence. The main novelty of this paper is creating a dictionary allowing one to transfer statements in Diophantine approximation in positive characteristic to combinatorics through the concept of a number wall, and conversely. |
| title | Combinatorics on Number Walls and the $P(t)$-adic Littlewood Conjecture |
| topic | Number Theory |
| url | https://arxiv.org/abs/2307.00955 |