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Register a new userOn the rational approximation to Thue--Morse rational numbers
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Let $b ge 2$ and $ell ge 1$ be integers. We establish that there is an absolute real number $K$ such that all the partial quotients of the rational number $$ prod_{h = 0}^ell , (1 - b^{-2^h}), $$ of denominator $b^{2^{ell+1} - 1}$, do not exceed $exp(K (log b)^2 sqrt{ell} 2^{ell/2})$.
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For a nonincreasing function $psi$, let $textrm{Exact}(psi)$ be the set of complex numbers that are approximable by complex rational numbers to order $psi$ but to no better order. In this paper, we obtain the Hausdorff dimension and packing dimension of $textrm{Exact}(psi)$ when $psi(x)=o(x^{-2})$. We also prove that the lower bound of the Hausdorff dimension is greater than $2-tau/(1-2tau)$ when $tau=limsup_{xtoinfty}psi(x)x^2$ small enough.
In this paper we present two efficient methods for reconstructing a rational number from several residue-modulus pairs, some of which may be incorrect. One method is a natural generalization of that presented by Wang, Guy and Davenport in cite{WGD1982} (for reconstructing a rational number from textit{correct} modular images), and also of an algorithm presented in cite{Abb1991} for reconstructing an textit{integer} value from several residue-modulus pairs, some of which may be incorrect.
In 1922, Mordell conjectured the striking statement that for a polynomial equation $f(x,y)=0$, if the topology of the set of complex number solutions is complicated enough, then the set of rational number solutions is finite. This was proved by Faltings in 1983, and again by a different method by Vojta in 1991, but neither proof provided a way to provably find all the rational solutions, so the search for other proofs has continued. Recently, Lawrence and Venkatesh found a third proof, relying on variation in families of $p$-adic Galois representations; this is the subject of the present exposition.
We provide in this paper an upper bound for the number of rational points on a curve defined over a one variable function field over a finite field. The bound only depends on the curve and the field, but not on the Jacobian variety of the curve.
Manins conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the Poisson formula. An alternative approach to Manins conjecture via universal torsors was used so far mainly over the field Q of rational numbers. In this note, we give a proof of Manins conjecture over the Gaussian rational numbers Q(i) and over other imaginary quadratic number fields with class number 1 for the singular toric cubic surface defined by t^3=xyz.
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