Let $X:=\mathbb{A}_{R}^{n}$ be the $n$-dimensional affine space over a discrete valuation ring $R$ with fraction field $K$. We prove that any pointed torsor $Y$ over $\mathbb{A}_{K}^{n}$ under the action of an affine finite-type group scheme can be extended to a torsor over $\mathbb{A}_{R}^{n}$ possibly after pulling $Y$ back over an automorphism of $\mathbb{A}_{K}^{n}$. The proof is effective. Other cases, including $X=\unicode[STIX]{x1D6FC}_{p,R}$, are also discussed.

In this article we obtain an explicit formula for certain Rankin–Selberg type Dirichlet series associated to certain Siegel cusp forms of half-integral weight. Here these Siegel cusp forms of half-integral weight are obtained from the composition of the Ikeda lift and the Eichler–Zagier–Ibukiyama correspondence. The integral weight version of the main theorem was obtained by Katsurada and Kawamura. The result of the integral weight case is a product of an $L$-function and Riemann zeta functions, while the half-integral weight case is an infinite summation over negative fundamental discriminants with certain infinite products. To calculate an explicit formula for such Rankin–Selberg type Dirichlet series, we use a generalized Maass relation and adjoint maps of index-shift maps of Jacobi forms.

We characterize all bounded Hankel operators $\unicode[STIX]{x1D6E4}$ such that $\unicode[STIX]{x1D6E4}^{\ast }\unicode[STIX]{x1D6E4}$ has finite spectrum. We identify spectral data corresponding to such operators and construct inverse spectral theory including the characterization of these spectral data.

In this paper we discuss some dimension results for triangle sets of compact sets in $\mathbb{R}^{2}$. In particular we prove that for any compact set $F$ in $\mathbb{R}^{2}$, the triangle set $\unicode[STIX]{x1D6E5}(F)$ satisfies

$$\begin{eqnarray}\dim _{\text{A}}\unicode[STIX]{x1D6E5}(F)\geqslant {\textstyle \frac{3}{2}}\dim _{\text{A}}F.\end{eqnarray}$$

$\dim _{\text{A}}F>1$

$$\begin{eqnarray}\dim _{\text{A}}\unicode[STIX]{x1D6E5}(F)\geqslant 1+\dim _{\text{A}}F.\end{eqnarray}$$

$\dim _{\text{A}}F>4/3$

$$\begin{eqnarray}\dim _{\text{A}}\unicode[STIX]{x1D6E5}(F)\geqslant \min \{{\textstyle \frac{5}{2}}\dim _{\text{A}}F-1,3\}.\end{eqnarray}$$

$F$

$$\begin{eqnarray}\text{}\underline{\dim _{\text{B}}}F\geqslant {\textstyle \frac{3}{2}}\text{}\underline{\dim _{\text{B}}}\unicode[STIX]{x1D6E5}(F)\quad \text{and}\quad \overline{\dim _{\text{B}}}F\geqslant {\textstyle \frac{3}{2}}\overline{\dim _{\text{B}}}\unicode[STIX]{x1D6E5}(F).\end{eqnarray}$$

We consider sequences of the form $(a_{n}\unicode[STIX]{x1D6FC})_{n}$ mod 1, where $\unicode[STIX]{x1D6FC}\in [0,1]$ and where $(a_{n})_{n}$ is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all $\unicode[STIX]{x1D6FC}$ in the sense of Lebesgue measure, we say that $(a_{n})_{n}$ has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of $(a_{n})_{n}$. Bloom, Chow, Gafni and Walker speculated that there might be a convergence/divergence criterion which fully characterizes the metric pair correlation property in terms of the additive energy, similar to Khintchine’s criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence $(a_{n})_{n}$ having large additive energy which, however, maintains the metric pair correlation property.

We aim to re-prove a theorem conjectured by Gauss, namely there are exactly nine imaginary quadratic fields $\mathbf{Q}(\sqrt{-q})$ with class number one: specifically the list is $q\in \{3,4,7,8,11,19,43,67,163\}$. Our method initially follows an idea of Goldfeld, but rather than using an elliptic curve of analytic rank three (provided by the Gross–Zagier theorem), we instead use an elliptic curve of analytic rank two, where this $L$-function vanishing can be proven by modular symbols rather than a difficult height formula. It is already clear that Goldfeld’s work yields a constant lower bound for the class number by such means, but unfortunately it seems that even for the best choice of elliptic curve this numerical constant is less than 1, unless one can show non-trivial cancellation in the $L$-function coefficients restricted to values taken by quadratic forms. To show the latter, we consider a specific analytic rank-two elliptic curve with complex multiplication by $\mathbf{Q}(\sqrt{-1})$, and then by adapting a result of Hooley’s regarding equi-distrbution of roots of a quadratic polynomial to varying moduli, are able to show that there is indeed sufficient coefficient cancellation, giving an effective resolution of class number one. As we use various aspects of the principal form, our proof seems inapplicable for larger class numbers. We also comment on the possibility of using spectral techniques (following Templier and Tsimerman) to show the desired coefficient cancellation, though postpone the details of this to elsewhere.

We apply Newton’s method to stochastic functional evolution equations in Hilbert spaces using semigroup methods. The first-order convergence is based on our generalization of the Gronwall-type inequality. We also establish a second-order convergence in a probabilistic sense.

Utilizing frameworks developed by Delsarte, Yudin and Levenshtein, we deduce linear programming lower bounds (as $N\rightarrow \infty$) for the Riesz energy of $N$-point configurations on the $d$-dimensional unit sphere in the so-called hypersingular case; i.e., for non-integrable Riesz kernels of the form $|x-y|^{-s}$ with $s>d$. As a consequence, we immediately get (thanks to the poppy-seed bagel theorem) lower estimates for the large $N$ limits of minimal hypersingular Riesz energy on compact $d$-rectifiable sets. Furthermore, for the Gaussian potential $\exp (-\unicode[STIX]{x1D6FC}|x-y|^{2})$ on $\mathbb{R}^{p}$, we obtain lower bounds for the energy of infinite configurations having a prescribed density.

We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known: they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. For all such spaces the best possible bounds for the quadratic discrepancies and sums of pairwise distances are obtained in the paper (Theorems 2.1 and 2.2). Distributions of points of $t$-designs on such spaces are also considered (Theorem 2.3). In particular, it is shown that the optimal $t$-designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances (Corollary 2.1). Our approach is based on the Fourier analysis on two-point homogeneous spaces and explicit spherical function expansions for discrepancies and sums of distances (Theorems 4.1 and 4.2).

We generalize a method of Conrey and Ghosh [Simple zeros of the Ramanujan $\unicode[STIX]{x1D70F}$-Dirichlet series. Invent. Math. 94(2) (1988), 403–419] to prove quantitative estimates for simple zeros of modular form $L$-functions of arbitrary conductor.

In 2005 Bourgain gave the first explicit construction of a two-source extractor family with min-entropy rate less than $1/2$. His approach combined Fourier analysis with innovative but inefficient tools from arithmetic combinatorics and yielded an unspecified min-entropy rate which was greater than $0.499$. This remained essentially the state of the art until a 2015 breakthrough by Chattopadhyay and Zuckerman in which they gave an alternative approach which produced extractors with arbitrarily small min-entropy rate. In the current work, we revisit the Fourier analytic approach. We give an improved analysis of one of Bourgain’s extractors which shows that it in fact extracts from sources with min-entropy rate near $\frac{21}{44}=0.477\ldots .$ Moreover, we construct a variant of this extractor which we show extracts from sources with min-entropy rate near $4/9=0.444\ldots .$ While this min-entropy rate is inferior to Chattopadhyay and Zuckerman’s construction, our extractors have the advantage of exponential small error, which is important in some applications. The key ingredient in these arguments is recent progress connected to the restriction theory of the finite field paraboloid by Rudnev and Shkredov. This in turn relies on a Rudnev’s point–plane incidence estimate, which in turn relies on Kollár’s generalization of the Guth–Katz incidence theorem.

We prove a range of new sum-product type growth estimates over a general field $\mathbb{F}$, in particular the special case $\mathbb{F}=\mathbb{F}_{p}$. They are unified by the theme of “breaking the $3/2$ threshold”, epitomizing the previous state of the art. This concerns two questions pivotal for the sum-product theory, which are lower bounds for the number of distinct cross-ratios determined by a finite subset of $\mathbb{F}$, as well as the number of values of the symplectic form determined by a finite subset of $\mathbb{F}^{2}$. We establish the estimate $|R[A]|\gtrsim |A|^{8/5}$ for cardinality of the set $R[A]$ of distinct cross-ratios, defined by triples of elements of a set $A\subset \mathbb{F}$ (sufficiently small if $\mathbb{F}$ has positive characteristic, similarly for the rest of the estimates), pinned at infinity. The cross-ratio bound enables us to break the threshold in the second question: for a non-collinear point set $P\subset \mathbb{F}^{2}$, the number of distinct values of the symplectic form $\unicode[STIX]{x1D714}$ on pairs of distinct points $u,u^{\prime }$ of $P$ is $|\unicode[STIX]{x1D714}(P)|\gtrsim |P|^{2/3+c}$, with an explicit $c$. Symmetries of the cross-ratio underlie its local growth properties under both addition and multiplication, yielding an onset of growth of products of difference sets, which is another main result herein. Our proofs make use of specially suited applications of new incidence bounds over $\mathbb{F}$, which apply to higher moments of representation functions. The technical thrust of the paper uses additive combinatorics to relate and adapt these higher moment bounds to growth estimates. A particular instance of this is breaking the threshold in the few sums, many products question over any $\mathbb{F}$, by showing that if $A$ is sufficiently small and has additive doubling constant $M$, then $|AA|\gtrsim M^{-2}|A|^{14/9}$. This result has a second moment version, which allows for new upper bounds for the number of collinear point triples in the set $A\times A\subset \mathbb{F}^{2}$, the quantity often arising in applications of geometric incidence estimates.

We establish Diophantine inequalities for the fractional parts of generalized polynomials, in particular for sequences $\unicode[STIX]{x1D708}(n)=\lfloor n^{c}\rfloor +n^{k}$ with $c>1$ a non-integral real number and $k\in \mathbb{N}$, as well as for $\unicode[STIX]{x1D708}(p)$ where $p$ runs through all prime numbers. This is related to classical work of Heilbronn and to recent results of Bergelson et al.

A flag area measure on an $n$-dimensional euclidean vector space is a continuous translation-invariant valuation with values in the space of signed measures on the flag manifold consisting of a unit vector $v$ and a $(p+1)$-dimensional linear subspace containing $v$ with $0\leqslant p\leqslant n-1$. Using local parallel sets, Hinderer constructed examples of $\text{SO}(n)$-covariant flag area measures. There is an explicit formula for his flag area measures evaluated on polytopes, which involves the squared cosine of the angle between two subspaces. We construct a more general sequence of smooth $\text{SO}(n)$-covariant flag area measures via integration over the normal cycle of appropriate differential forms. We provide an explicit description of our measures on polytopes, which involves an arbitrary elementary symmetric polynomial in the squared cosines of the principal angles between two subspaces. Moreover, we show that these flag area measures span the space of all smooth $\text{SO}(n)$-covariant flag area measures, which gives a classification result in the spirit of Hadwiger’s theorem.

We study the fine-scale $L^{2}$-mass distribution of toral Laplace eigenfunctions with respect to random position in two and three dimensions. In two dimensions, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established in the optimal Planck-scale regime. In three dimensions the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain’s eigenfunctions”). Other than the said precise results, lower and upper bounds are proved for the variance under more general flatness assumptions on the Fourier coefficients.

Following Wigert, various authors, including Ramanujan, Gronwall, Erdős, Ivić, Schwarz, Wirsing and Shiu, determined the maximal order of several multiplicative functions, generalizing Wigert’s result

$$\begin{eqnarray}\max _{n\leqslant x}\log d(n)=\frac{\log x}{\log \log x}(\log 2+o(1)).\end{eqnarray}$$

$\log f(f(n))$

$f$

$f$

$r_{2}$

$$\begin{eqnarray}\max _{n\leqslant x}\log r_{2}(r_{2}(n))=\frac{\sqrt{\log x}}{\log \log x}(c/\sqrt{2}+o(1))\end{eqnarray}$$

$c>0$