We consider finite point subsets (distributions) in compact metric spaces. In the case of general rectifiable metric spaces, non-trivial bounds for sums of distances between points of distributions and for discrepancies of distributions in metric balls are given (Theorem 1.1). We generalize Stolarsky’s invariance principle to distance-invariant spaces (Theorem 2.1). For arbitrary metric spaces, we prove a probabilistic invariance principle (Theorem 3.1). Furthermore, we construct equal-measure partitions of general rectifiable compact metric spaces into parts of small average diameter (Theorem 4.1).

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.

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

$$\begin{eqnarray}\Vert f(p)\Vert <p^{-\unicode[STIX]{x1D70E}}\end{eqnarray}$$

$p$

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

We introduce a new arithmetic function $a(n)$ defined to be the number of random multiplications by residues modulo $n$ before the running product is congruent to zero modulo $n$ . We give several formulas for computing the values of this function and analyze its asymptotic behavior. We find that it is closely related to $P_{1}(n)$ , the largest prime divisor of $n$ . In particular, $a(n)$ and $P_{1}(n)$ have the same average order asymptotically. Furthermore, the difference between the functions $a(n)$ and $P_{1}(n)$ is $o(1)$ as $n$ tends to infinity on a set with density approximately $0.623$ . On the other hand, however, we see that (except on a set of density zero) the difference between $a(n)$ and $P_{1}(n)$ tends to infinity on the integers outside this set. Finally, we consider the asymptotic behavior of the difference between these two functions and find that $\sum _{n\leqslant x}(a(n)-P_{1}(n))\sim (1-\unicode[STIX]{x1D70B}/4)\sum _{n\leqslant x}P_{2}(n)$ , where $P_{2}(n)$ is the second largest divisor of $n$ .

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 $K$ 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 prove the analog of Cramér’s short intervals theorem for primes in arithmetic progressions and prime ideals, under the relevant Riemann hypothesis. Both results are uniform in the data of the underlying structure. Our approach is based mainly on the inertia property of the counting functions of primes and prime ideals.

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 $S$ .

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

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

$c>0$

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

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

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 $(q,a_{1}a_{2})=1$ .

In this paper, we propose a conjectural identity between the Fourier–Jacobi periods on symplectic groups and the central value of certain Rankin–Selberg $L$ -functions. This identity can be viewed as a refinement to the global Gan–Gross–Prasad conjecture for $\text{Sp}(2n)\times \text{Mp}(2m)$ . To support this conjectural identity, we show that when $n=m$ and $n=m\pm 1$ , it can be deduced from the Ichino–Ikeda conjecture in some cases via theta correspondences. As a corollary, the conjectural identity holds when $n=m=1$ or when $n=2$ , $m=1$ and the automorphic representation on the bigger group is endoscopic.

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.

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.

By constructing suitable Borcherds forms on Shimura curves and using Schofer’s formula for norms of values of Borcherds forms at CM points, we determine all of the equations of hyperelliptic Shimura curves $X_{0}^{D}(N)$ . As a byproduct, we also address the problem of whether a modular form on Shimura curves $X_{0}^{D}(N)/W_{D,N}$ with a divisor supported on CM divisors can be realized as a Borcherds form, where $X_{0}^{D}(N)/W_{D,N}$ denotes the quotient of $X_{0}^{D}(N)$ by all of the Atkin–Lehner involutions. The construction of Borcherds forms is done by solving certain integer programming problems.

Let $h(d)$ be the class number of indefinite binary quadratic forms of discriminant $d$ and let $\unicode[STIX]{x1D700}_{d}$ be the corresponding fundamental unit. In this paper, we obtain an asymptotic formula for the $k$ th moment of $h(d)$ over positive discriminants $d$ with $\unicode[STIX]{x1D700}_{d}\leqslant x$ , uniformly for real numbers $k$ in the range $0<k\leqslant (\log x)^{1-o(1)}$ . This improves upon the work of Raulf, who obtained such an asymptotic for a fixed positive integer $k$ . We also investigate the distribution of large values of $h(d)$ when the discriminants $d$ are ordered according to the size of their fundamental units $\unicode[STIX]{x1D700}_{d}$ . In particular, we show that the tail of this distribution has the same shape as that of class numbers of imaginary quadratic fields ordered by the size of their discriminants. As an application of these results, we prove that there are many positive discriminants $d$ with class number $h(d)\geqslant (e^{\unicode[STIX]{x1D6FE}}/3+o(1))\cdot \unicode[STIX]{x1D700}_{d}(\log \log \unicode[STIX]{x1D700}_{d})/\log \,\unicode[STIX]{x1D700}_{d}$ , a bound that we believe is best possible. We also obtain an upper bound for $h(d)$ that is twice as large, assuming the generalized Riemann hypothesis.

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

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

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

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

$1\leqslant r\leqslant k-1$

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

$k\geqslant 3$

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

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

This paper is motivated by Davenport’s problem and the subsequent work regarding badly approximable points in submanifolds of a Euclidean space. We study the problem in the area of twisted Diophantine approximation and present two different approaches. The first approach shows that, under a certain restriction, any countable intersection of the sets of weighted badly approximable points on any non-degenerate ${\mathcal{C}}^{1}$ submanifold of $\mathbb{R}^{n}$ has full dimension. In the second approach, we introduce the property of isotropically winning and show that the sets of weighted badly approximable points are isotropically winning under the same restriction as above.

We prove an asymptotic formula for squarefree numbers in arithmetic progressions, improving previous results by Prachar and Hooley. As a consequence we improve a lower bound of Heath-Brown for the least squarefree number in an arithmetic progression.

We improve a recent result of B. Hanson [Estimates for character sums with various convolutions. Preprint, 2015, arXiv:1509.04354] on multiplicative character sums with expressions of the type $a+b+cd$ and variables $a,b,c,d$ from four distinct sets of a finite field. We also consider similar sums with $a+b(c+d)$ . Our new bounds rely on some recent advances in additive combinatorics.