Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Approximation to an extremal number, its square and its cube   We study rational approximation properties for successive powers of extremal
numbers defined by Roy. For $n\in{\{1,2\}}$, the classic approximation
constants
$\lambda_{n}(\zeta),\hat{\lambda}_{n}(\zeta),w_{n}(\zeta),\hat{w}_{n}(\zeta)$
connected to an extremal number $\zeta$ have been established and in fact much
more is known. However, so far almost nothing had been known for $n\geq 3$. In
this paper we determine all classic approximation constants as above for $n=3$.
Our methods will more generally provide detailed information on the combined
graph defined by Schmidt and Summerer assigned to an extremal number, its
square and its cube. We provide some results for $n=4$ as well. In the course
of the proof of the main results we establish a very general connection between
Khintchine's transference inequalities and uniform approximation.

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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
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The categories of this paper are: supr-con
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The categories of this paper are: econ
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The categories of this paper are: hep-ph
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The categories of this paper are: q-fin
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The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
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The categories of this paper are: plasm-ph
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The categories of this paper are: eess
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The categories of this paper are: quant-ph