Consequences of the Continuity of the Monic Integer Transfinite Diameter

Jan Hilmar

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Consequences of the Continuity of the Monic Integer Transfinite Diameter

Jan Hilmar

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Abstract

We consider the problem of determining the monic integer transfinite diameter for real intervals $I$ of length less than 4. We show that $t_M([0,x])$, as a function in $x>0$, is continuous, therefore disproving two conjectures due to Hare and Smyth. Consequently, for $n>2\in\naturals$, we define the quantity $b_{\max}(n)&=&\sup_{b>\frac{1}{n}}\left\{b|t_M([0,b])=\tfrac{1}{n}\right.\right\}$ and give lower and upper bounds of $b_{\max}(n)$. Finally, we improve the lower bound for $b_{\max}(n)$ for $3\leq n\leq 8$.

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Jan Hilmar. 2007-06-06. Consequences of the Continuity of the Monic Integer Transfinite Diameter. https://arxiv.org/abs/math/0703888

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