Related papers: Diophantine approximation with integers having no …

For $\lambda \in (1/2, 1)$ and $\alpha$, we consider sets of numbers $x$ such that for infinitely many $n$, $x$ is $2^{-\alpha n}$-close to some $\sum_{i=1}^n \omega_i \lambda^i$, where $\omega_i \in \{0,1\}$. These sets are in Falconer's…

Given $\boldsymbol{\alpha} \in [0,1]^d$, we estimate the smooth discrepancy of the Kronecker sequence $(n \boldsymbol{\alpha} \,\mathrm{mod}\, 1)_{n\geq 1}$. We find that it can be smaller than the classical discrepancy of $\textbf{any}$…

We investigate exponential sums over those numbers $\leq x$ all of whose prime factors are $\leq y$. We prove fairly good minor arc estimates, valid whenever $\log^{3}x \leq y \leq x^{1/3}$. Then we prove sharp upper bounds for the $p$-th…

For fixed integer $a\ge3$, we study the binary Diophantine equation $\frac{a}n=\frac1x+\frac1y$ and in particular the number $E_a(N)$ of $n\le N$ for which the equation has no positive integer solutions in $x, y$. The asymptotic formula…

An integer is said to be $y$-friable if its greatest prime factor P(n) is less than $y$. In this paper, we study numbers of the shape $n-1$ when $P(n)\leq y$ and $n\leq x$. One expects that, statistically, their multiplicative behaviour…

Let $\alpha$ be an irrational real number. We show that the set of $\epsilon$-badly approximable numbers \[ \mathrm{Bad}^\varepsilon (\alpha) := \{x\in [0,1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| q\alpha -x \| \geq \varepsilon \} \]…

Let $[\,\cdot\,]$ be the floor function. In this paper we show that whenever $\eta$ is real, the constants $\lambda_i$ satisfy some necessary conditions, then for any fixed $1<c<38/37$ there exist infinitely many prime triples $p_1,\,…

By applying inter-universal Teichm\"uller theory and its slight modification over the rational number field, we prove new Diophantine results towards effective abc inequalities and the generalized Fermat equations. For coprime integers $a,…

We elaborate on a problem raised by Schmidt in 1967 which generalizes the theory of classical Diophantine approximation to subspaces of $\R^n$. We consider Diophantine exponents for linear subspaces of $\R^n$ which generalize the…

Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is…

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 $|x-a/q|<\Delta_q$.…

For a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd…

We investigate lower bounds for the variance in arithmetic progressions of certain multiplicative functions "close" to $1$. Specifically, we consider $\alpha_N$-fold divisor functions, when $\alpha_N$ is a sequence of positive real numbers…

A nontrivial solution of the equation A!B! = C! is a triple of positive integers (A, B, C) with A $\le$ B $\le$ C -- 2. It is conjectured that the only nontrivial solution is (6, 7, 10), and this conjecture has been checked up to C = 10 6.…

Let $1<c<832/825$. For large real numbers $N>0$ and a small constant $\vartheta>0$, the inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\vartheta \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3,\,p_4$ such that,…

We prove a generalization of W.M. Schmidt's theorem related to the Diophantine approximations for a linear form of the type $\alpha_1x_1+\alpha_2x_2 +y$ with {\it positive} integers $x_1,x_2$.

The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $\nu>0$ one can establish the infinitude…

In this paper, we use a variety of classical and new research methods for ternary exponential Diophantine equations and extensive use of computer calculations to study the conjecture of R. Scott and R. Styer which asserts that for any fixed…

There is a large literature on the asymptotic distribution of numbers free of large prime factors, so-called $\textit{smooth}$ or $\textit{friable}$ numbers. But there is very little known about this distribution that is numerically…

Let $\sigma_{i}(x_{1},\ldots, x_{n})=\sum_{1\leq k_{1}<k_{2}<\ldots <k_{i}\leq n}x_{k_{1}}\ldots x_{k_{i}}$ be the $i$-th elementary symmetric polynomial. In this note we generalize and extend the results obtained in a recent work of Zhang…