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Register a new userThe difference between the Hurwitz continued fraction expansions of a complex number and its rational approximations
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For regular continued fraction, if a real number $x$ and its rational approximation $p/q$ satisfying $|x-p/q|<1/q^2$, then, after deleting the last integer of the partial quotients of $p/q$, the sequence of the remaining partial quotients is a prefix of that of $x$. In this paper, we show that the situation is completely different if we consider the Hurwitz continued fraction expansions of a complex number and its rational approximations. More specifically, we consider the set $E(psi)$ of complex numbers which are well approximated with the given bound $psi$ and have quite different Hurwitz continued fraction expansions from that of their rational approximations. The Hausdorff and packing dimensions of such set are determined. It turns out that its packing dimension is always full for any given approximation bound $psi$ and its Hausdorff dimension is equal to that of the $psi$-approximable set $W(psi)$ of complex numbers. As a consequence, we also obtain an analogue of the classical Jarnik Theorem in real case.
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Let $q>2$ be an odd integer. For each integer $x$ with $0<x<q$ and $(q,x)= 1$, we know that there exists one and only one $bar{x}$ with $0<bar{x}<q$ such that $xbar{x}equiv1(bmod q)$. A Lehmer number is defined to be any integer $a$ with $2dagger(a+bar{a})$. For any nonnegative integer $k$, Let $$ M(x,q,k)=displaystylemathop {displaystylemathop{sum{}}_{a=1}^{q} displaystylemathop{sum{}}_{bleq xq}}_{mbox{$tinybegin{array}{c} 2|a+b+1 abequiv1(bmod q)end{array}$}}(a-b)^{2k}.$$ The main purpose of this paper is to study the properties of $M(x,q,k)$, and give a sharp asymptotic formula, by using estimates of Kloostermans sums and properties of trigonometric sums.
For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article ``Uniform Bounds on Pre-Images Under Quadratic Dynamical Systems, by two of the present authors and five others, it was shown that the number of rational iterated pre-images of the origin is bounded as one varies the morphism in a certain one-dimensional family. Subject to the validity of the Birch and Swinnerton-Dyer conjecture and some other related conjectures for the L-series of a specific abelian variety and using a number of modern tools for locating rational points on high genus curves, we show that the maximum number of rational iterated pre-images is six. We also provide further insight into the geometry of the ``pre-image curves.
The goal of this paper is to formulate a systematical method for constructing the fastest possible continued fraction approximations of a class of functions. The main tools are the multiple-correction method, the generalized Morticis lemma and the Mortici-transformation. As applications, we will present some sharp inequalities, and the continued fraction expansions associated to the volume of the unit ball. In addition, we obtain a new continued fraction expansion of Ramanujan for a ratio of the gamma functions, which is showed to be the fastest possible. Finally, three conjectures are proposed.
We exhibit a method to use continued fractions in function fields to find new families of hyperelliptic curves over the rationals with given torsion order in their Jacobians. To show the utility of the method, we exhibit a new infinite family of curves over $mathbb Q$ with genus two whose Jacobians have torsion order eleven.
In this short note we prove two elegant generalized continued fraction formulae $$e= 2+cfrac{1}{1+cfrac{1}{2+cfrac{2}{3+cfrac{3}{4+ddots}}}}$$ and $$e= 3+cfrac{-1}{4+cfrac{-2}{5+cfrac{-3}{6+cfrac{-4}{7+ddots}}}}$$ using elementary methods. The first formula is well-known, and the second one is newly-discovered in arXiv:1907.00205 [cs.LG]. We then explore the possibility of automatic verification of such formulae using computer algebra systems (CASs).
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