In this paper we study the Duffin–Schaeffer conjecture, which claims that $\unicode[STIX]{x1D706}(\bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }{\mathcal{E}}_{n})=1$ if and only if $\sum _{n=1}^{\infty }\unicode[STIX]{x1D706}({\mathcal{E}}_{n})=\infty$ , where $\unicode[STIX]{x1D706}$ denotes the Lebesgue measure on $\mathbb{R}/\mathbb{Z}$ ,

$$\begin{eqnarray}{\mathcal{E}}_{n}={\mathcal{E}}_{n}(\unicode[STIX]{x1D713})=\mathop{\bigcup }_{\substack{ m=1 \\ (m,n)=1}}^{n}\bigg(\frac{m-\unicode[STIX]{x1D713}(n)}{n},\frac{m+\unicode[STIX]{x1D713}(n)}{n}\bigg),\end{eqnarray}$$

$\unicode[STIX]{x1D713}$

$\bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }{\mathcal{E}}_{n}$

$\bigcup _{n=1}^{\infty }{\mathcal{E}}_{n}$

$C>0$

$$\begin{eqnarray}\unicode[STIX]{x1D706}\bigg(\mathop{\bigcup }_{n=1}^{\infty }{\mathcal{E}}_{n}\bigg)\geqslant C\min \bigg\{\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D706}({\mathcal{E}}_{n}),1\bigg\}.\end{eqnarray}$$

$p$

For a half-integral weight modular form $f=\sum _{n=1}^{\infty }a_{f}(n)n^{(k-1)/2}q^{n}$ of weight $k=\ell +1/2$ on $\unicode[STIX]{x1D6E4}_{0}(4)$ such that $a_{f}(n)$ ( $n\in \mathbb{N}$ ) are real, we prove for a fixed suitable natural number $r$ that $a_{f}(n)$ changes sign infinitely often as $n$ varies over numbers having at most $r$ prime factors, assuming the analog of the Ramanujan conjecture for Fourier coefficients of half-integral weight forms.

A $(d-1)$-dimensional simplicial complex is called balanced if its underlying graph admits a proper $d$-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs). Specifically, we prove the balanced analog of the celebrated lower bound theorem (LBT) for normal pseudomanifolds and characterize the case of equality; we introduce and characterize the balanced analog of the Walkup class; and we propose the balanced analog of the generalized lower bound conjecture (GLBC) and establish some related results. We close with constructions of balanced manifolds with few vertices.

Dyadic shifts $D\oplus T$ of point distributions $D$ in the $d$-dimensional unit cube $U^{d}$ are considered as a form of randomization. Explicit formulas for the $L_{q}$-discrepancies of such randomized distributions are given in the paper in terms of Rademacher functions. Relying on the statistical independence of Rademacher functions, Khinchin’s inequalities, and other related results, we obtain very sharp upper and lower bounds for the mean $L_{q}$-discrepancies, $0<q\leqslant \infty$. The upper bounds imply directly a generalization of the well-known Chen theorem on mean discrepancies with respect to dyadic shifts (Theorem 2.1). From the lower bounds, it follows that for an arbitrary $N$-point distribution $D_{N}$ and any exponent $0<q\leqslant 1$, there exist dyadic shifts $D_{N}\oplus T$ such that the $L_{q}$-discrepancy ${\mathcal{L}}_{q}[D_{N}\oplus T]>c_{d,q}(\log N)^{(1/2)(d-1)}$ (Theorem 2.2). The lower bounds for the $L_{\infty }$-discrepancy are also considered in the paper. It is shown that for an arbitrary $N$-point distribution $D_{N}$, there exist dyadic shifts $D_{N}\oplus T$ such that ${\mathcal{L}}_{\infty }[D_{N}\oplus T]>c_{d}(\log N)^{(1/2)d}$ (Theorem 2.3).

An important result of Weyl states that for every sequence $(a_{n})_{n\geqslant 1}$ of distinct positive integers the sequence of fractional parts of $(a_{n}{\it\alpha})_{n\geqslant 1}$ is uniformly distributed modulo one for almost all ${\it\alpha}$. However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy of $(\{a_{n}{\it\alpha}\})_{n\geqslant 1}$ for almost all ${\it\alpha}$. In particular, it is very difficult to give sharp lower bounds for the speed of convergence. Until now this was only carried out for lacunary sequences $(a_{n})_{n\geqslant 1}$ and for some special cases such as the Kronecker sequence $(\{n{\it\alpha}\})_{n\geqslant 1}$ or the sequence $(\{n^{2}{\it\alpha}\})_{n\geqslant 1}$. In the present paper we answer the question for a large class of sequences $(a_{n})_{n\geqslant 1}$ including as a special case all polynomials $a_{n}=P(n)$ with $P\in \mathbb{Z}[x]$ of degree at least 2.

Laplace operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of types ${\it\delta}$ and ${\it\delta}^{\prime }$. Assuming rational independence of edge lengths, necessary and sufficient conditions for isospectrality of two Laplacians defined on the same graph are derived and scrutinized. It is proved that the spectrum of a graph Laplacian uniquely determines matching conditions for “almost all” graphs.

We relate Catalan numbers and Catalan determinants to random matrix integrals and to moments of spin representations of odd orthogonal groups.

It is proved that the resolvent norm of an operator with a compact resolvent on a Banach space $X$ cannot be constant on an open set if the underlying space or its dual is complex strictly convex. It is also shown that this is not the case for an arbitrary Banach space: there exists a separable, reflexive space $X$ and an unbounded, densely defined operator acting in $X$ with a compact resolvent whose norm is constant in a neighbourhood of zero; moreover $X$ is isometric to a Hilbert space on a subspace of co-dimension $2$. There is also a bounded linear operator acting on the same space whose resolvent norm is constant in a neighbourhood of zero. It is shown that similar examples cannot exist in the co-dimension $1$ case.

We prove that the log-Brunn–Minkowski inequality (log-BMI) for the Lebesgue measure in dimension $n$ would imply the log-BMI and, therefore, theB-conjecture for any even log-concave measure in dimension $n$ . As a consequence, we prove the log-BMI and the B-conjecture for any even log-concave measure in the plane. Moreover, we prove that the log-BMI reduces to the following: for each dimension $n$ , there is a density $f_{n}$ , which satisfies an integrability assumption, so that the log-BMI holds for parallelepipeds with parallel facets, for the density $f_{n}$ . As a byproduct of our methods, we study possible log-concavity of the function $t\mapsto |(K+_{p}\cdot ~\text{e}^{t}L)^{\circ }|$ , where $p\geqslant 1$ and $K$ , $L$ are symmetric convex bodies, which we are able to prove in some instances and, as a further application, we confirm the variance conjecture in a special class of convex bodies. Finally, we establish a non-trivial dual form of the log-BMI.

An orthogonal coloring of the two-dimensional unit sphere $\mathbb{S}^{2}$, is a partition of $\mathbb{S}^{2}$ into parts such that no part contains a pair of orthogonal points: that is, a pair of points at spherical distance ${\it\pi}/2$ apart. It is a well-known result that an orthogonal coloring of $\mathbb{S}^{2}$ requires at least four parts, and orthogonal colorings with exactly four parts can easily be constructed from a regular octahedron centered at the origin. An intriguing question is whether or not every orthogonal 4-coloring of $\mathbb{S}^{2}$ is such an octahedral coloring. In this paper, we address this question and show that if every color class has a non-empty interior, then the coloring is octahedral. Some related results are also given.

In this paper we establish concavity properties of two extensions of the classical notion of the outer parallel volume. On the one hand, we replace the Lebesgue measure by more general measures. On the other hand, we consider a functional version of the outer parallel sets.

We show that substantially more than a quarter of the odd integers of the form $pq$ up to $x$, with $p,q$ both prime, satisfy $p\equiv q\equiv 3~(\text{mod}\,4)$.

We give non-trivial bounds for the bilinear sums

$$\begin{eqnarray}\mathop{\sum }_{u=1}^{U}\mathop{\sum }_{v=1}^{V}\unicode[STIX]{x1D6FC}_{u}\unicode[STIX]{x1D6FD}_{v}\,\mathbf{e}_{p}(u/f(v)),\end{eqnarray}$$

$\,\mathbf{e}_{p}(z)$

$\mathbb{F}_{p}$

$p$

$U$

$V$

$f\in \mathbb{F}_{p}[X]$

$\{\unicode[STIX]{x1D6FC}_{u}\}$

$\{\unicode[STIX]{x1D6FD}_{v}\}$

$f(X)=aX+b$

$\mathbb{F}_{p}$

In this paper, we study how small a box contains at least two points from a modular hyperbola $xy\equiv c\;(\text{mod}\;p)$ . There are two such points in a square of side length $p^{1/4+\unicode[STIX]{x1D716}}$ . Furthermore, it turns out that either there are two such points in a square of side length $p^{1/6+\unicode[STIX]{x1D716}}$ or the least quadratic non-residue is less than $p^{1/(6\sqrt{e})+\unicode[STIX]{x1D716}}$ .

Let $K$ be a convex body in $\mathbb{R}^{d}$ which slides freely in a ball. Let $K^{(n)}$ denote the intersection of $n$ closed half-spaces containing $K$ whose bounding hyperplanes are independent and identically distributed according to a certain prescribed probability distribution. We prove an asymptotic formula for the expectation of the difference of the volumes of $K^{(n)}$ and $K$, and an asymptotic upper bound on the variance of the volume of $K^{(n)}$. We obtain these asymptotic formulas by proving results for weighted mean width approximations of convex bodies that admit a rolling ball by inscribed random polytopes and then using polar duality to convert them into statements about circumscribed random polytopes.

Let ${\rm\Lambda}(n)$ be the von Mangoldt function, $x$ be real and $2\leqslant y\leqslant x$. This paper improves the estimate for the exponential sum over primes in short intervals

$$\begin{eqnarray}S_{k}(x,y;{\it\alpha})=\mathop{\sum }_{x<n\leqslant x+y}{\rm\Lambda}(n)e(n^{k}{\it\alpha})\end{eqnarray}$$

$k\geqslant 3$

${\it\alpha}$

$n$

$s$

$k$

In this paper, we investigate in various ways the representation of a large natural number as a sum of a fixed power of Piatetski-Shapiro numbers, thereby establishing a variant of the Hilbert–Waring problem with numbers from a sparse sequence.

We establish lower bounds for (i) the numbers of positive and negative terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.

The isotropic constant $L_{K}$ is an affine-invariant measure of the spread of a convex body $K$. For a $d$-dimensional convex body $K$, $L_{K}$ can be defined by $L_{K}^{2d}=\det (A(K))/(\text{vol}(K))^{2}$, where $A(K)$ is the covariance matrix of the uniform distribution on $K$. It is an open problem to find a tight asymptotic upper bound of the isotropic constant as a function of the dimension. It has been conjectured that there is a universal constant upper bound. The conjecture is known to be true for several families of bodies, in particular, highly symmetric bodies such as bodies having an unconditional basis. It is also known that maximizers cannot be smooth. In this work we study bodies that are neither smooth nor highly symmetric by showing progress towards reducing to a highly symmetric case among non-smooth bodies. More precisely, we study the set of maximizers among simplicial polytopes and we show that if a simplicial polytope $K$ is a maximizer of the isotropic constant among $d$-dimensional convex bodies, then when $K$ is put in isotropic position it is symmetric around any hyperplane spanned by a $(d-2)$-dimensional face and the origin. By a result of Campi, Colesanti and Gronchi, this implies that a simplicial polytope that maximizes the isotropic constant must be a simplex.

Let $Q(x,y,z)$ be an integral quadratic form with determinant coprime to some modulus $q$. We show that $q\,|\,Q$ for some non-zero integer vector $(x,y,z)$ of length $O(q^{5/8+{\it\varepsilon}})$, for any fixed ${\it\varepsilon}>0$. Without the coprimality condition on the determinant one could not necessarily achieve an exponent below $2/3$. The proof uses a bound for short character sums involving binary quadratic forms, which extends a result of Chang.