We consider the classical theta operator ${\it\theta}$ on modular forms modulo $p^{m}$ and level $N$ prime to $p$, where $p$ is a prime greater than three. Our main result is that ${\it\theta}$ mod $p^{m}$ will map forms of weight $k$ to forms of weight $k+2+2p^{m-1}(p-1)$ and that this weight is optimal in certain cases when $m$ is at least two. Thus, the natural expectation that ${\it\theta}$ mod $p^{m}$ should map to weight $k+2+p^{m-1}(p-1)$ is shown to be false. The primary motivation for this study is that application of the ${\it\theta}$ operator on eigenforms mod $p^{m}$ corresponds to twisting the attached Galois representations with the cyclotomic character. Our construction of the ${\it\theta}$-operator mod $p^{m}$ gives an explicit weight bound on the twist of a modular mod $p^{m}$ Galois representation by the cyclotomic character.

For any number field we calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field.

Let $E(N)$ denote the number of positive integers $n\leqslant N$, with $n\equiv 4\;(\text{mod}\;24)$, which cannot be represented as the sum of four squares of primes. We establish that $E(N)\ll N^{11/32}$, thus improving on an earlier result of Harman and the first author, where the exponent $7/20$ appears in place of $11/32$.

Let $q\in (1,2)$. A $q$-expansion of a number $x$ in $[0,1/(q-1)]$ is a sequence $({\it\delta}_{i})_{i=1}^{\infty }\in \{0,1\}^{\mathbb{N}}$ satisfying

$$\begin{eqnarray}x=\mathop{\sum }_{i=1}^{\infty }\frac{{\it\delta}_{i}}{q^{i}}.\end{eqnarray}$$

${\mathcal{B}}_{\aleph _{0}}$

$q$

$x$

$q$

${\mathcal{B}}_{1,\aleph _{0}}$

$q$

$1$

$q$

$1=\sum _{i=1}^{\infty }q^{-n_{i}}$

$\min {\mathcal{B}}_{\aleph _{0}}=\min {\mathcal{B}}_{1,\aleph _{0}}=(1+\sqrt{5})/2$

${\mathcal{B}}_{\aleph _{0}}\cap ((1+\sqrt{5})/2,q_{1}]=\{q_{1}\}$

$q_{1}\,({\approx}1.64541)$

$x^{6}-x^{4}-x^{3}-2x^{2}-x-1=0$

${\mathcal{B}}_{1,\aleph _{0}}$

$q_{3}\,({\approx}1.68042)$

$x^{5}-x^{4}-x^{3}-x+1=0$

${\mathcal{B}}_{\aleph _{0}}\cap (q_{1},q_{3}]=\{q_{2},q_{3}\}$

$q_{2}\,({\approx}1.65462)$

$x^{6}-2x^{4}-x^{3}-1=0$

We investigate the problem of the decomposition of balls into a finite number of congruent pieces in dimension $d=2k$. In addition, we prove that the $d$-dimensional unit ball $B_{d}$ can be divided into a finite number of congruent pieces if $d=4$ or $d\geqslant 6$. We show that the minimal number of required pieces is less than $20d$, if $d\geqslant 10$.

We study the meromorphic continuation and the spectral expansion of the opposite-sign Kloosterman sum zeta function

$$\begin{eqnarray}\displaystyle (2{\it\pi}\sqrt{mn})^{2s-1}\mathop{\sum }_{\ell =1}^{\infty }\frac{S(m,-n,\ell )}{\ell ^{2s}} & & \displaystyle \nonumber\end{eqnarray}$$

$m,n$

$s\in \mathbb{C}$

Let $C\in \mathbb{Z}[x_{1},\ldots ,x_{n}]$ be a cubic form. Assume that $C$ splits into four forms. Then $C(x_{1},\ldots ,x_{n})=0$ has a non-trivial integer solution provided that $n\geqslant 10$.

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.

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.

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.

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 $p\equiv q\equiv 3~(\text{mod}\,4)$.

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.

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.

In this paper we show that sieve methods used previously to investigate primes in short intervals and corresponding Goldbach-type problems can be modified to obtain results on primes in Beatty sequences in short intervals.

We show that if a finite, large enough subset $A$ of an arbitrary abelian group $G$ satisfies the small doubling condition $|A+A|\leqslant (\log |A|)^{1-{\it\varepsilon}}|A|$, then $A$ must contain a three-term arithmetic progression whose terms are not all equal, and $A+A$ must contain an arithmetic progression or a coset of a subgroup, either of which is of size at least $\exp [c(\log |A|)^{{\it\delta}}]$. This extends analogous results obtained by Sanders, and by Croot, Łaba and Sisask in the case where $G=\mathbb{Z}$.

We present a construction of a measure-zero Kakeya-type set in a finite-dimensional space $K^{n}$ over a local field with finite residue field. The construction is an adaptation of the ideas appearing in works by Sawyer [Mathematika34(1) (1987), 69–76] and Wisewell [Mathematika51(1–2) (2004), 155–162]. The existence of measure-zero Kakeya-type sets over discrete valuation rings is also discussed, giving an alternative construction to the one over $\mathbb{F}_{\ell }\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$ presented by Dummit and Hablicsek [Mathematika59(2) (2013), 257–266].

The illumination problem may be phrased as the problem of covering a convex body in Euclidean $n$-space by a minimum number of translates of its interior. By a probabilistic argument, we show that, arbitrarily close to the Euclidean ball, there is a centrally symmetric convex body of illumination number exponentially large in the dimension.