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arXiv · 1708.09146

A combinatorial approach to the Littlewood conjecture in a field of formal series

Sanghoon Kwon
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Abstract

A long-standing conjecture of Littlewood about simultaneous Diophantine approximation has an analogous problem for a field of formal Laurent series $\mathbb{F}(\!(t^{-1})\!)$. That is, we can ask whether for any series $Θ$, $Φ$ and any $ε>0$, there is a polynomial $α$ such that $|α|\langleαΘ\rangle\langleαΦ\rangle$ where $\langleΘ\rangle=\underset{β\in\mathbb{F}[t]}{\inf}|Θ-β|$. If the base field $\mathbb{F}$ is infinite, then the answer is negative due to Davenport and Lewis (1963). We give a connection between the combinatorics of an orbit under a semigroup action and Diophantine approximation problem when $\mathbb{F}$ is finite.

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Sanghoon Kwon. 2019-02-26. A combinatorial approach to the Littlewood conjecture in a field of formal series. https://arxiv.org/abs/1708.09146

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