相关论文: Best rational approximations of an irrational numb…
While the theory of operator approximation with any given accuracy is well elaborated, the theory of {best constrained} constructive operator approximation is still not so well developed. Despite increasing demands from applications this…
In this paper we develop a new explicit method to studying rational points near manifolds and obtain optimal lower bounds on the number of rational points of bounded height lying at a given distance from an arbitrary non-degenerate curve.…
We study approximation in the unit interval by rational numbers whose numerators are selected randomly with certain probabilities. Previous work showed that an analogue of Khintchine's Theorem holds in a similar random model and raised the…
We present a new rational approximation algorithm based on the empirical interpolation method for interpolating a family of parametrized functions to rational polynomials with invariant poles, leading to efficient numerical algorithms for…
In this paper, we present a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts…
It is well-known that the second-order cone can be outer-approximated to an arbitrary accuracy $\epsilon$ by a polyhedral cone of compact size defined by irrational data. In this paper, we propose two rational polyhedral…
We present a new algorithm for reconstructing an exact algebraic number from its approximate value using an improved parameterized integer relation construction method. Our result is consistent with the existence of error controlling on…
Given real numbers whose sum is an integer, we study the problem of finding integers which match these real numbers as closely as possible, in the sense of L^p norm, while preserving the sum. We describe the structure of solutions for this…
Inspired by a problem proposed by Mahler, we will address the following related question, 'How well can irrationals in a missing digit set be approximated by rationals with polynomial denominators?' and prove some related results. To…
The infinite pigeonhole principle for $k$ colors ($\mathsf{RT}_k$) states, for every $k$-partition $A_0 \sqcup \dots \sqcup A_{k-1} = \mathbb{N}$, the existence of an infinite subset~$H \subseteq A_i$ for some~$i < k$. This seemingly…
Given quantities $\Delta_1,\Delta_2,\dots\geqslant 0$, a fundamental problem in Diophantine approximation is to understand which irrational numbers $x$ have infinitely many reduced rational approximations $a/q$ such that $|x-a/q|<\Delta_q$.…
By using a jump transformation associated to the Romik map, we define a new continued fraction algorithm called odd-odd continued fraction, whose principal convergents are rational numbers of odd denominators and odd numerators. Among…
We study how well a real number can be approximated by sums of two or more rational numbers with denominators up to a certain size.
There are numerous ways to represent real numbers. We may use, e.g., Cauchy sequences, Dedekind cuts, numerical base-10 expansions, numerical base-2 expansions and continued fractions. If we work with full Turing computability, all these…
We give a polynomial-time constant-factor approximation algorithm for maximum independent set for (axis-aligned) rectangles in the plane. Using a polynomial-time algorithm, the best approximation factor previously known is $O(\log\log n)$.…
We study the close connection between rational functions that approximate a given Boolean function, and quantum algorithms that compute the same function using postselection. We show that the minimal degree of the former equals (up to a…
We give an easy optimal bound for the dimension of the subspaces generated by the best Diophantine approximations.
This study reexamines diffusive representations for fractional integrals with the goal of pioneering new variants of such representations. These variants aim to offer highly efficient numerical algorithms for the approximate computation of…
We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in…
The papers shows an algorithm to search for approximations of reals to rationals of the form a/b^2 that runs on \sqrt(b) polynomial time steps.