Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Connection Between Irrationality Measures and Polynomial
  Continued Fractions   Linear recursions with integer coefficients, such as the one generating the
Fibonacci sequence, have been intensely studied over millennia and yet still
hide new mathematics. Such a recursion was used by Ap\'ery in his proof of the
irrationality of $\zeta(3)$, later named the Ap\'ery constant. Ap\'ery's proof
used a specific linear recursion containing integer polynomials forming a
continued fraction; called polynomial continued fractions (PCFs). Similar
polynomial recursions prove the irrationality of other mathematical constants
such as $\pi$ and $e$. More generally, the sequences generated by PCFs form
Diophantine approximations (DAs), which are ubiquitous in areas of math such as
number theory. It is not known which polynomial recursions create useful DAs
and whether they prove irrationality. Here, we present general conclusions and
conjectures about DAs created from PCFs. Specifically, we generalize Ap\'ery's
work, going beyond his particular choice of PCF, finding the conditions under
which a PCF proves irrationality or provides an efficient DA. To provide
concrete examples, we apply our findings to PCFs found by the Ramanujan Machine
algorithms to represent fundamental constants such as $\pi$, $e$, $\zeta(3)$,
and the Catalan constant G. For each such PCF, we demonstrate the extraction of
its convergence rate and efficiency, as well as the bound it provides for the
irrationality measure of the fundamental constant. We further propose new DA
conjectures based on PCFs. Our findings motivate future research on sequences
created by any linear recursions with integer coefficients, to aid the
development of systematic algorithms for finding DAs of fundamental constants.
Consequently, our study may contribute to ongoing efforts to answer open
questions, such as the proof of the irrationality of the Catalan constant or of
values of the Riemann zeta function (e.g., $\zeta(5)$).

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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-ex
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The categories of this paper are: econ
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The categories of this paper are: alg-geom
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The categories of this paper are: funct-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-th
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The categories of this paper are: comp-gas
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The categories of this paper are: cs
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-lat