Employing the inverse function theorem on Banach spaces, we prove that in a $C^{2}(S^{n-1})$ -neighborhood of the unit ball, the only solutions of $\unicode[STIX]{x1D6F1}^{2}K=cK$ are origin-centered ellipsoids. Here $K$ is an $n$ -dimensional convex body, $\unicode[STIX]{x1D6F1}K$ is the projection body of $K$ and $\unicode[STIX]{x1D6F1}^{2}K=\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D6F1}K).$

We first show that every algebraic torus over any field, not necessarily split, can be realized as the special fiber of a semi-abelian scheme whose generic fiber is an absolutely simple abelian variety. Then we investigate which algebraic tori can be thus obtained, when we require the generic fiber of the semi-abelian scheme to carry non-trivial endomorphism structures.

In this paper we provide a unified approach, based on methods of descriptive set theory, for proving some classical selection theorems. Among them is the zero-dimensional Michael selection theorem, the Kuratowski–Ryll-Nardzewski selection theorem, as well as a known selection theorem for hyperspaces.

Let $s(\cdot )$ denote the sum-of-proper-divisors function, that is, $s(n)=\sum _{d\mid n,~d<n}d$ . Erdős, Granville, Pomerance, and Spiro conjectured that for any set $\mathscr{A}$ of asymptotic density zero, the preimage set $s^{-1}(\mathscr{A})$ also has density zero. We prove a weak form of this conjecture: if $\unicode[STIX]{x1D716}(x)$ is any function tending to $0$ as $x\rightarrow \infty$ , and $\mathscr{A}$ is a set of integers of cardinality at most $x^{1/2+\unicode[STIX]{x1D716}(x)}$ , then the number of integers $n\leqslant x$ with $s(n)\in \mathscr{A}$ is $o(x)$ , as $x\rightarrow \infty$ . In particular, the EGPS conjecture holds for infinite sets with counting function $O(x^{1/2+\unicode[STIX]{x1D716}(x)})$ . We also disprove a hypothesis from the same paper of EGPS by showing that for any positive numbers $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D716}$ , there are integers $n$ with arbitrarily many $s$ -preimages lying between $\unicode[STIX]{x1D6FC}(1-\unicode[STIX]{x1D716})n$ and $\unicode[STIX]{x1D6FC}(1+\unicode[STIX]{x1D716})n$ . Finally, we make some remarks on solutions $n$ to congruences of the form $\unicode[STIX]{x1D70E}(n)\equiv a~(\text{mod}~n)$ , proposing a modification of a conjecture appearing in recent work of the first two authors. We also improve a previous upper bound for the number of solutions $n\leqslant x$ , making it uniform in $a$ .

We study the Goldbach problem for primes represented by the polynomial $x^{2}+y^{2}+1$ . The set of such primes is sparse in the set of all primes, but the infinitude of such primes was established by Linnik. We prove that almost all even integers $n$ satisfying certain necessary local conditions are representable as the sum of two primes of the form $x^{2}+y^{2}+1$ . This improves a result of Matomäki, which tells us that almost all even $n$ satisfying a local condition are the sum of one prime of the form $x^{2}+y^{2}+1$ and one generic prime. We also solve the analogous ternary Goldbach problem, stating that every large odd $n$ is the sum of three primes represented by our polynomial. As a byproduct of the proof, we show that the primes of the form $x^{2}+y^{2}+1$ contain infinitely many three-term arithmetic progressions, and that the numbers $\unicode[STIX]{x1D6FC}p~(\text{mod}~1)$ , with $\unicode[STIX]{x1D6FC}$ irrational and $p$ running through primes of the form $x^{2}+y^{2}+1$ , are distributed rather uniformly.

We study the minimal gap statistic for fractional parts of sequences of the form ${\mathcal{A}}^{\unicode[STIX]{x1D6FC}}=\{\unicode[STIX]{x1D6FC}a(n)\}$, where ${\mathcal{A}}=\{a(n)\}$ is a sequence of distinct integers. Assuming that the additive energy of the sequence is close to its minimal possible value, we show that for almost all $\unicode[STIX]{x1D6FC}$, the minimal gap $\unicode[STIX]{x1D6FF}_{\min }^{\unicode[STIX]{x1D6FC}}(N)=\min \{\unicode[STIX]{x1D6FC}a(m)-\unicode[STIX]{x1D6FC}a(n)\hspace{0.2em}{\rm mod}\hspace{0.2em}1:1\leqslant m\neq n\leqslant N\}$ is close to that of a random sequence.

Barannikov and Kontsevich [Frobenius manifolds and formality of Lie algebras of polyvector fields. Int. Math. Res. Not. IMRN1998(4) (1998), 201–215], constructed a DGBV (differential Gerstenhaber–Batalin–Vilkovisky) algebra $\mathbf{t}$ for a compact smooth Calabi–Yau complex manifold $M$ of dimension $m$, which gives rise to the $B$-side formal Frobenius manifold structure in the homological mirror symmetry conjecture. The cohomology of the DGBV algebra $\mathbf{t}$ is isomorphic to the total singular cohomology $H^{\bullet }(M)=\bigoplus _{k=0}^{2m}H^{k}(M,\mathbb{C})$ of $M$. If $M=X_{G}(\mathbb{C})$, where $X_{G}$ is the hypersurface defined by a homogeneous polynomial $G(\text{}\underline{x})$ in the projective space $\mathbb{P}^{n}$, then we give a purely algorithmic construction of a DGBV algebra ${\mathcal{A}}_{U}$, which computes the primitive part $\bigoplus _{k=0}^{m}\mathbf{PH}^{k}$ of the middle-dimensional cohomology $\bigoplus _{k=0}^{m}H^{k}(M,\mathbb{C})$, using the de Rham cohomology of the hypersurface complement $U_{G}:=\mathbb{P}^{n}\setminus X_{G}$ and the residue isomorphism from $H_{\text{dR}}^{k}(U_{G}/\mathbb{C})$ to $\mathbf{PH}^{k}$. We observe that the DGBV algebra ${\mathcal{A}}_{U}$ still makes sense even for a singular projective Calabi–Yau hypersurface, i.e. ${\mathcal{A}}_{U}$ computes $\bigoplus _{k=0}^{m}H_{\text{dR}}^{k}(U_{G}/\mathbb{C})$ even for a singular $X_{G}$. Moreover, we give a precise relationship between ${\mathcal{A}}_{U}$ and $\mathbf{t}$ when $X_{G}$ is smooth in $\mathbf{P}^{n}$.

If $3\leqslant n<\unicode[STIX]{x1D714}$ and $V$ is a vector space over $\mathbb{Q}$ with $|V|\leqslant \aleph _{n-2}$ , then there is a well ordering of $V$ such that every vector is the last element of only finitely many length- $n$ arithmetic progressions ( $n$ -APs). This implies that there is a set mapping $f:V\rightarrow [V]^{{<}\unicode[STIX]{x1D714}}$ with no free set which is an $n$ -AP. If, however, $|V|\geqslant \aleph _{n-1}$ , then for every set mapping $f:V\rightarrow [V]^{{<}\unicode[STIX]{x1D714}}$ there is a free set which is an $n$ -AP.

Let $P^{+}(n)$ denote the largest prime factor of the integer $n$ and $P_{y}^{+}(n)$ denote the largest prime factor $p$ of $n$ which satisfies $p\leqslant y$ . In this paper, first we show that the triple consecutive integers with the two patterns $P^{+}(n-1)>P^{+}(n)<P^{+}(n+1)$ and $P^{+}(n-1)<P^{+}(n)>P^{+}(n+1)$ have a positive proportion respectively. More generally, with the same methods we can prove that for any$J\in \mathbb{Z}$ , $J\geqslant 3$ , the $J$ -tuple consecutive integers with the two patterns $P^{+}(n+j_{0})=\min _{0\leqslant j\leqslant J-1}P^{+}(n+j)$ and $P^{+}(n+j_{0})=\max _{0\leqslant j\leqslant J-1}P^{+}(n+j)$ also have a positive proportion, respectively. Second, for $y=x^{\unicode[STIX]{x1D703}}$ with $0<\unicode[STIX]{x1D703}\leqslant 1$ we show that there exists a positive proportion of integers $n$ such that $P_{y}^{+}(n)<P_{y}^{+}(n+1)$ . Specifically, we can prove that the proportion of integers $n$ such that $P^{+}(n)<P^{+}(n+1)$ is larger than 0.1356, which improves the previous result “0.1063” of the author.

The purpose of this paper twofold. Firstly, we establish $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness and $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness of several different classes of multifractal decomposition sets of arbitrary Borel measures (satisfying a mild non-degeneracy condition and two mild “smoothness” conditions). Secondly, we apply these results to study the $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness and $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$ -completeness of several multifractal decomposition sets of self-similar measures (satisfying a mild separation condition). For example, a corollary of our results shows if $\unicode[STIX]{x1D707}$ is a self-similar measure satisfying the strong separation condition and $\unicode[STIX]{x1D707}$ is not equal to the normalized Hausdorff measure on its support, then the classical multifractal decomposition sets of $\unicode[STIX]{x1D707}$ defined by

$$\begin{eqnarray}\bigg\{x\in \mathbb{R}^{d}\,\bigg|\,\lim _{r{\searrow}0}\frac{\log \unicode[STIX]{x1D707}(B(x,r))}{\log r}=\unicode[STIX]{x1D6FC}\bigg\}\end{eqnarray}$$

$\unicode[STIX]{x1D6F1}_{3}^{0}$

We show that integral monodromy groups of Kloosterman $\ell$-adic sheaves of rank $n\geqslant 2$ on $\mathbb{G}_{m}/\mathbb{F}_{q}$ are as large as possible when the characteristic $\ell$ is large enough, depending only on the rank. This variant of Katz’s results over $\mathbb{C}$ was known by works of Gabber, Larsen, Nori and Hall under restrictions such as $\ell$ large enough depending on $\operatorname{char}(\mathbb{F}_{q})$ with an ineffective constant, which is unsuitable for applications. We use the theory of finite groups of Lie type to extend Katz’s ideas, in particular the classification of maximal subgroups of Aschbacher and Kleidman–Liebeck. These results will apply to study hyper-Kloosterman sums and their reductions in forthcoming work.

For the superelliptic curves of the form

$$\begin{eqnarray}(x+1)\cdots (x+i-1)(x+i+1)\cdots (x+k)=y^{\ell }\end{eqnarray}$$

$y\neq 0$

$k\geqslant 3$

$\ell \geqslant 2,$

$i\in [2,k]\setminus \unicode[STIX]{x1D6FA}$

$\ell <\text{e}^{3^{k}}.$

$\unicode[STIX]{x1D6FA}$

$[p_{\unicode[STIX]{x1D703}},(k-p_{\unicode[STIX]{x1D703}}))$

$p_{\unicode[STIX]{x1D703}}$

$k/2$

$i=1$

In 2006 Brown asked the following question in the spirit of Ramsey theory: given a non-periodic infinite word $x=x_{1}x_{2}x_{3}\ldots$ with values in a set $\mathbb{A}$ , does there exist a finite colouring $\unicode[STIX]{x1D711}:\mathbb{A}^{+}\rightarrow C$ relative to which $x$ does not admit a $\unicode[STIX]{x1D711}$ -monochromatic factorization, i.e. a factorization of the form $x=u_{1}u_{2}u_{3}\ldots$ with $\unicode[STIX]{x1D711}(u_{i})=\unicode[STIX]{x1D711}(u_{\!j})$ for all $i,j\geqslant 1$ ? Various partial results in support of an affirmative answer to this question have appeared in the literature in recent years. In particular it is known that the question admits an affirmative answer for all non-uniformly recurrent words and for various classes of uniformly recurrent words including Sturmian words and fixed points of strongly recognizable primitive substitutions. In this paper we give a complete and optimal affirmative answer to this question by showing that if $x=x_{1}x_{2}x_{3}\ldots$ is an infinite non-periodic word with values in a set $\mathbb{A}$ , then there exists a $2$ -colouring $\unicode[STIX]{x1D711}:\mathbb{A}^{+}\rightarrow \{0,1\}$ such that for any factorization $x=u_{1}u_{2}u_{3}\ldots$ we have $\unicode[STIX]{x1D711}(u_{i})\neq \unicode[STIX]{x1D711}(u_{\!j})$ for some $i\neq j$ .

Let $A$ be a set of natural numbers. Recent work has suggested a strong link between the additive energy of $A$ (the number of solutions to $a_{1}+a_{2}=a_{3}+a_{4}$ with $a_{i}\in A$) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of $A$ modulo $1$. There appears to be reasonable evidence to speculate a sharp Khinchin-type threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the convergence theory, in other words the extent to which having a low additive energy forces a set to be metric Poissonian.

We find logarithmic asymptotics of $L_{2}$ -small deviation probabilities for weighted stationary Gaussian processes (both for real and complex-valued) having a power-type discrete or continuous spectrum. Our results are based on the spectral theory of pseudo-differential operators developed by Birman and Solomyak.

This paper is devoted to dimensional reductions via the norm-resolvent convergence. We derive explicit bounds on the resolvent difference as well as spectral asymptotics. The efficiency of our abstract tool is demonstrated by its application on seemingly different partial differential equation problems from various areas of mathematical physics; all are analysed in a unified manner, known results are recovered and new ones established.

We prove that the exponent of distribution of $\unicode[STIX]{x1D70F}_{3}$ in arithmetic progressions can be as large as $\frac{1}{2}+\frac{1}{34}$, provided that the moduli is squarefree and has only sufficiently small prime factors. The tools involve arithmetic exponent pairs for algebraic trace functions, as well as a double $q$-analogue of the van der Corput method for smooth bilinear forms.

For a non-empty polyhedral set $P\subset \mathbb{R}^{d}$ , let ${\mathcal{F}}(P)$ denote the set of faces of $P$ , and let $N(P,F)$ be the normal cone of $P$ at the non-empty face $F\in {\mathcal{F}}(P)$ . We prove the identity

$$\begin{eqnarray}\mathop{\sum }_{F\in {\mathcal{F}}(P)}(-1)^{\operatorname{dim}F}\unicode[STIX]{x1D7D9}_{F-N(P,F)}=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }P\text{ is bounded},\\ 0\quad & \text{if }P\text{ is unbounded and line-free}.\end{array}\right.\end{eqnarray}$$

$0$

We prove that for any positive integers $k,n$ with $n>\frac{3}{2}(k^{2}+k+2)$ , prime $p$ , and integers $c,a_{i}$ , with $p\nmid a_{i}$ , $1\leqslant i\leqslant n$ , there exists a solution $\text{}\underline{x}$ to the congruence

$$\begin{eqnarray}\mathop{\sum }_{i=1}^{n}a_{i}x_{i}^{k}\equiv c\hspace{0.6em}({\rm mod}\hspace{0.2em}p)\end{eqnarray}$$

$1\leqslant {x_{i}\ll }_{k}p^{1/k}$

$1\leqslant i\leqslant n$

$n$

$\unicode[STIX]{x1D700}>0$

$c_{\unicode[STIX]{x1D700}}$

$n>c_{\unicode[STIX]{x1D700}}k$

$1\leqslant {x_{i}\ll }_{\unicode[STIX]{x1D700},k}p^{1/k+\unicode[STIX]{x1D700}}$

We investigate the approximation of quadratic Dirichlet $L$ -functions over function fields by truncations of their Euler products. We first establish representations for such $L$ -functions as products over prime polynomials times products over their zeros. This is the hybrid formula in function fields. We then prove that partial Euler products are good approximations of an $L$ -function away from its zeros and that, when the length of the product tends to infinity, we recover the original $L$ -function. We also obtain explicit expressions for the arguments of quadratic Dirichlet $L$ -functions over function fields and for the arguments of their partial Euler products. In the second part of the paper we construct, for each quadratic Dirichlet $L$ -function over a function field, an auxiliary function based on the approximate functional equation that equals the $L$ -function on the critical line. We also construct a parametrized family of approximations of these auxiliary functions and prove that the Riemann hypothesis holds for them and that their zeros are related to those of the associated $L$ -function. Finally, we estimate the counting function for the zeros of this family of approximations, show that these zeros cluster near those of the associated $L$ -function, and that, when the parameter is not too large, almost all the zeros of the approximations are simple.