Mathematika

Let $f$ be a polynomial of degree $k>1$ with irrational leading coefficient. We obtain results of the form

$$\begin{eqnarray}\Vert f(p)\Vert

$p$

$\unicode[STIX]{x1D6FC}p^{k}$

We prove a dimension-free strong uniformity theorem, and apply it in the configuration space of a large system of non-interacting particles, to describe the fast approach to equilibrium starting from off-equilibrium, and its long-term stability.

Given $n\in \mathbb{N}$ and $\unicode[STIX]{x1D70F}>1/n$ , let ${\mathcal{S}}_{n}(\unicode[STIX]{x1D70F})$ denote the classical set of $\unicode[STIX]{x1D70F}$ -approximable points in $\mathbb{R}^{n}$ , which consists of $\mathbf{x}\in \mathbb{R}^{n}$ that lie within distance $q^{-\unicode[STIX]{x1D70F}-1}$ from the lattice $(1/q)\mathbb{Z}^{n}$ for infinitely many $q\in \mathbb{N}$ . In pioneering work, Kleinbock and Margulis showed that for any non-degenerate submanifold ${\mathcal{M}}$ of $\mathbb{R}^{n}$ and any $\unicode[STIX]{x1D70F}>1/n$ almost all points on ${\mathcal{M}}$ are not $\unicode[STIX]{x1D70F}$ -approximable. Numerous subsequent papers have been geared towards strengthening this result through investigating the Hausdorff measure and dimension of the associated null set ${\mathcal{M}}\cap {\mathcal{S}}_{n}(\unicode[STIX]{x1D70F})$ . In this paper we suggest a new approach based on the Mass Transference Principle of Beresnevich and Velani [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], which enables us to find a sharp lower bound for $\dim {\mathcal{M}}\cap {\mathcal{S}}_{n}(\unicode[STIX]{x1D70F})$ for any $C^{2}$ submanifold ${\mathcal{M}}$ of $\mathbb{R}^{n}$ and any $\unicode[STIX]{x1D70F}$ satisfying $1/n\leqslant \unicode[STIX]{x1D70F}<1/m$ . Here $m$ is the codimension of ${\mathcal{M}}$ . We also show that the condition on $\unicode[STIX]{x1D70F}$ is best possible and extend the result to general approximating functions.

A strong quantitative form of Manin’s conjecture is established for a certain variety in biprojective space. The singular integral in an application of the circle method involves the third power of the integral sine function and is evaluated in closed form.

Let $I_{s,k,r}(X)$ denote the number of integral solutions of the modified Vinogradov system of equations

$$\begin{eqnarray}x_{1}^{j}+\cdots +x_{s}^{j}=y_{1}^{j}+\cdots +y_{s}^{j}\quad (1\leqslant j\leqslant k,\;j\neq r),\end{eqnarray}$$

$1\leqslant x_{i},y_{i}\leqslant X\;(1\leqslant i\leqslant s)$

$I_{s,k,r}(X)$

$1\leqslant r\leqslant k-1$

$s,k\in \mathbb{N}$

$k\geqslant 3$

$1\leqslant s\leqslant (k^{2}-1)/2$

$I_{s,k,1}(X)\ll X^{s+\unicode[STIX]{x1D700}}$

Building on recent work of Bhargava, Elkies and Schnidman and of Kriz and Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.

Let $S=\{q_{1},\ldots ,q_{s}\}$ be a finite, non-empty set of distinct prime numbers. For a non-zero integer $m$ , write $m=q_{1}^{r_{1}}\cdots q_{s}^{r_{s}}M$ , where $r_{1},\ldots ,r_{s}$ are non-negative integers and $M$ is an integer relatively prime to $q_{1}\cdots q_{s}$ . We define the $S$ -part $[m]_{S}$ of $m$ by $[m]_{S}:=q_{1}^{r_{1}}\cdots q_{s}^{r_{s}}$ . Let $(u_{n})_{n\geqslant 0}$ be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every $\unicode[STIX]{x1D700}>0$ , there exists an integer $n_{0}$ such that $[u_{n}]_{S}\leqslant |u_{n}|^{\unicode[STIX]{x1D700}}$ holds for $n>n_{0}$ . Our proof is ineffective in the sense that it does not give an explicit value for $n_{0}$ . Under various assumptions on $(u_{n})_{n\geqslant 0}$ , we also give effective, but weaker, upper bounds for $[u_{n}]_{S}$ of the form $|u_{n}|^{1-c}$ , where $c$ is positive and depends only on $(u_{n})_{n\geqslant 0}$ and $S$ .

There are several formulas in classical prime number theory that are said to be “equivalent” to the Prime Number Theorem. For Beurling generalized numbers, not all such implications hold unconditionally. Here we investigate conditions under which the Beurling version of these relations do or do not hold.

The discrepancy function measures the deviation of the empirical distribution of a point set in $[0,1]^{d}$ from the uniform distribution. In this paper, we study the classical discrepancy function with respect to the bounded mean oscillation and exponential Orlicz norms, as well as Sobolev, Besov and Triebel–Lizorkin norms with dominating mixed smoothness. We give sharp bounds for the discrepancy function under such norms with respect to infinite sequences.

We show that smooth-supported multiplicative functions $f$ are well distributed in arithmetic progressions $a_{1}a_{2}^{-1}\;(\text{mod}~q)$ on average over moduli $q\leqslant x^{3/5-\unicode[STIX]{x1D700}}$ with $(q,a_{1}a_{2})=1$ .

The multiplicative group generated by a certain sequence of rationals is determined, settling a 30-year conjecture.

Define $r_{4}(N)$ to be the largest cardinality of a set $A\subset \{1,\ldots ,N\}$ that does not contain four elements in arithmetic progression. In 1998, Gowers proved that

$$\begin{eqnarray}r_{4}(N)\ll N(\log \log N)^{-c}\end{eqnarray}$$

$c>0$

$$\begin{eqnarray}r_{4}(N)\ll N\text{e}^{-c\sqrt{\log \log N}}.\end{eqnarray}$$

$$\begin{eqnarray}r_{4}(N)\ll N(\log N)^{-c},\end{eqnarray}$$

For a finite field of odd cardinality $q$ , we show that the sequence of iterates of $aX^{2}+c$ , starting at $0$ , always recurs after $O(q/\text{log}\log q)$ steps. For $X^{2}+1$ , the same is true for any starting value. We suggest that the traditional “birthday paradox” model is inappropriate for iterates of $X^{3}+c$ , when $q$ is 2 mod 3.

Let $K$ be an algebraic number field of degree $d\geqslant 3$ , $\unicode[STIX]{x1D70E}_{1},\unicode[STIX]{x1D70E}_{2},\ldots ,\unicode[STIX]{x1D70E}_{d}$ the embeddings of $K$ into $\mathbb{C}$ , $\unicode[STIX]{x1D6FC}$ a non-zero element in $K$ , $a_{0}\in \mathbb{Z}$ , $a_{0}>0$ and

$$\begin{eqnarray}F_{0}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC})Y).\end{eqnarray}$$

$\unicode[STIX]{x1D710}$

$K$

$a\in \mathbb{Z}$

$F_{0}(X,Y)\in \mathbb{Z}[X,Y]$

$\unicode[STIX]{x1D710}^{a}$

$a\in \mathbb{Z}$

$\unicode[STIX]{x1D710}$

$$\begin{eqnarray}F_{a}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})Y).\end{eqnarray}$$

$m>0$

$(x,y,a)\in \mathbb{Z}^{3}$

$$\begin{eqnarray}0<|F_{a}(x,y)|\leqslant m\end{eqnarray}$$

$xy\not =0$

$\mathbb{Q}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})=K$

$m$

$K$

$F_{0}$

$\unicode[STIX]{x1D710}$

$d$

We consider an arithmetic function defined independently by John G. Thompson and Greg Simay, with particular attention to its mean value, its maximal size, and the analytic nature of its Dirichlet series generating function.

Combinatorial discrepancy is a complexity measure of a collection of sets which quantifies how well the sets in the collection can be simultaneously balanced. More precisely, we are given an $n$ -point set $P$ , and a collection ${\mathcal{F}}=\{F_{1},\ldots ,F_{m}\}$ of subsets of $P$ , and our goal is color $P$ with two colors, red and blue, so that the maximum over the $F_{i}$ of the absolute difference between the number of red elements and the number of blue elements (the discrepancy) is minimized. Combinatorial discrepancy has many applications in mathematics and computer science, including constructions of uniformly distributed point sets, and lower bounds for data structures and private data analysis algorithms. We investigate the combinatorial discrepancy of geometrically defined systems, in which $P$ is an $n$ -point set in $d$ -dimensional space, and ${\mathcal{F}}$ is the collection of subsets of $P$ induced by dilations and translations of a fixed convex polytope $B$ . Such set systems include systems of sets induced by axis-aligned boxes, whose discrepancy is the subject of the well-known Tusnády problem. We prove new discrepancy upper and lower bounds for such set systems by extending the approach based on factorization norms previously used by the author, Matoušek, and Talwar. We also outline applications of our results to geometric discrepancy, data structure lower bounds, and differential privacy.

We transpose the parametric geometry of numbers, recently created by Schmidt and Summerer, to fields of rational functions in one variable and analyze, in that context, the problem of simultaneous approximation to exponential functions.

Introduced in Schmidt and Summerer [Parametric geometry of numbers and applications. Acta Arith. 140 (2009), 67–91], approximation exponents $\text{}\underline{\unicode[STIX]{x1D711}}_{i},\overline{\unicode[STIX]{x1D711}}_{i}$ , $(i=1,2,3)$ , attached to points $\boldsymbol{\unicode[STIX]{x1D709}}=(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2})$ in $\mathbb{R}^{2}$ , give information on Diophantine approximation properties of these points. Laurent [Exponents of Diophantine approximation in dimension two. Canad. J. Math. 61 (2009), 165–189] had described all possible quadruples $(\text{}\underline{\unicode[STIX]{x1D711}}_{1},\overline{\unicode[STIX]{x1D711}}_{1},\text{}\underline{\unicode[STIX]{x1D711}}_{3},\overline{\unicode[STIX]{x1D711}}_{3})$ arising in this way. Our emphasis here will be on $\text{}\underline{\unicode[STIX]{x1D711}}_{2},\overline{\unicode[STIX]{x1D711}}_{2}$ and the construction of suitable “ $3$ -systems”.