Gallagher’s theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.

We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based, not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves.

This paper is concerned with the maximisation of the $k$-th eigenvalue of the Laplacian amongst flat tori of unit volume in dimension $d$ as $k$ goes to infinity. We show that in any dimension maximisers exist for any given $k$, but that any sequence of maximisers degenerates as $k$ goes to infinity when the dimension is at most 10. Furthermore, we obtain specific upper and lower bounds for the injectivity radius of any sequence of maximisers. We also prove that flat Klein bottles maximising the $k$-th eigenvalue of the Laplacian exhibit the same behaviour. These results contrast with those obtained recently by Gittins and Larson, stating that sequences of optimal cuboids for either Dirichlet or Neumann boundary conditions converge to the cube no matter the dimension. We obtain these results via Weyl asymptotics with explicit control of the remainder in terms of the injectivity radius. We reduce the problem at hand to counting lattice points inside anisotropically expanding domains, where we generalise methods of Yu. Kordyukov and A. Yakovlev by considering domains that expand at different rates in various directions.

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.

We study a combinatorial problem that recently arose in the context of shape optimization: among all triangles with vertices $(0,0)$, $(x,0)$, and $(0,y)$ and fixed area, which one encloses the most lattice points from $\mathbb{Z}_{{>}0}^{2}$? Moreover, does its shape necessarily converge to the isosceles triangle $(x=y)$ as the area becomes large? Laugesen and Liu suggested that, in contrast to similar problems, there might not be a limiting shape. We prove that the limiting set is indeed non-trivial and contains infinitely many elements. We also show that there exist “bad” areas where no triangle is particularly good at capturing lattice points and show that there exists an infinite set of slopes $y/x$ such that any associated triangle captures more lattice points than any other fixed triangle for infinitely many (and arbitrarily large) areas; this set of slopes is a fractal subset of $[1/3,3]$ and has Minkowski dimension of at most $3/4$.

We generalize Skriganov’s notion of weak admissibility for lattices to include standard lattices occurring in Diophantine approximation and algebraic number theory, and we prove estimates for the number of lattice points in sets such as aligned boxes. Our result improves on Skriganov’s celebrated counting result if the box is sufficiently distorted, the lattice is not admissible, and, e.g., symplectic or orthogonal. We establish a criterion under which our error term is sharp, and we provide examples in dimensions $2$ and $3$ using continued fractions. We also establish a similar counting result for primitive lattice points, and apply the latter to the classical problem of Diophantine approximation with primitive points as studied by Chalk, Erdős, and others. Finally, we use o-minimality to describe large classes of sets to which our counting results apply.

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.

Introduced in Schmidt and Summerer [Parametric geometry of numbers and applications. Acta Arith.140 (2009), 67–91], approximation exponents $\text{}\underline{\unicode[STIX]{x1D711}}_{i},\overline{\unicode[STIX]{x1D711}}_{i}$, $(i=1,2,3)$, attached to points $\boldsymbol{\unicode[STIX]{x1D709}}=(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2})$ in $\mathbb{R}^{2}$, give information on Diophantine approximation properties of these points. Laurent [Exponents of Diophantine approximation in dimension two. Canad. J. Math.61 (2009), 165–189] had described all possible quadruples $(\text{}\underline{\unicode[STIX]{x1D711}}_{1},\overline{\unicode[STIX]{x1D711}}_{1},\text{}\underline{\unicode[STIX]{x1D711}}_{3},\overline{\unicode[STIX]{x1D711}}_{3})$ arising in this way. Our emphasis here will be on $\text{}\underline{\unicode[STIX]{x1D711}}_{2},\overline{\unicode[STIX]{x1D711}}_{2}$ and the construction of suitable “$3$-systems”.

Lattice sieving is asymptotically the fastest approach for solving the shortest vector problem (SVP) on Euclidean lattices. All known sieving algorithms for solving the SVP require space which (heuristically) grows as $2^{0.2075n+o(n)}$, where $n$ is the lattice dimension. In high dimensions, the memory requirement becomes a limiting factor for running these algorithms, making them uncompetitive with enumeration algorithms, despite their superior asymptotic time complexity.

We generalize sieving algorithms to solve SVP with less memory. We consider reductions of tuples of vectors rather than pairs of vectors as existing sieve algorithms do. For triples, we estimate that the space requirement scales as $2^{0.1887n+o(n)}$. The naive algorithm for this triple sieve runs in time $2^{0.5661n+o(n)}$. With appropriate filtering of pairs, we reduce the time complexity to $2^{0.4812n+o(n)}$ while keeping the same space complexity. We further analyze the effects of using larger tuples for reduction, and conjecture how this provides a continuous trade-off between the memory-intensive sieving and the asymptotically slower enumeration.

One of the most fruitful results from Minkowski’s geometric viewpoint on number theory is his so-called first fundamental theorem. It provides an optimal upper bound for the volume of a $0$ -symmetric convex body whose only interior lattice point is the origin. Minkowski also obtained a discrete analog by proving optimal upper bounds on the number of lattice points in the boundary of such convex bodies. Whereas the volume inequality has been generalized to any number of interior lattice points already by van der Corput in the 1930s, a corresponding result for the discrete case remained to be proven. Our main contribution is a corresponding optimal relation between the number of boundary and interior lattice points of a $0$ -symmetric convex body. The proof relies on a congruence argument and a difference set estimate from additive combinatorics.

We show that for any coprime integers λ1, . . ., λk and any finite A ⊂ $\mathbb{Z}$, one has

|\lambda_1 \cdot A + \cdots + \lambda_k \cdot A| \geq (|\lambda_1| + \cdots + |\lambda_k|)|A|- C,

The approximation constant ${\it\lambda}_{k}({\it\zeta})$ is defined as the supremum of ${\it\eta}\in \mathbb{R}$ such that the estimate $\max _{1\leqslant j\leqslant k}\Vert {\it\zeta}^{j}x\Vert \leqslant x^{-{\it\eta}}$ has infinitely many integer solutions $x$. Here $\Vert .\Vert$ denotes the distance to the closest integer. We establish a connection on the joint spectrum $({\it\lambda}_{1}({\it\zeta}),{\it\lambda}_{2}({\it\zeta}),\ldots )$, which will lead to various improvements of known results on the individual spectrum of the approximation constants ${\it\lambda}_{k}({\it\zeta})$ as well. In particular, for given $k\geqslant 1$ and ${\it\lambda}\geqslant 1$, we construct ${\it\zeta}$ in the Cantor set with ${\it\lambda}_{k}({\it\zeta})={\it\lambda}$. Moreover, we establish an estimate for the uniform approximation constants $\widehat{{\it\lambda}}_{k}({\it\zeta})$, which enables us to determine classical approximation constants for Liouville numbers.

The Frobenius number $F(\boldsymbol{a})$ of a lattice point $\boldsymbol{a}$ in $\mathbb{R}^{d}$ with positive coprime coordinates, is the largest integer which can not be expressed as a non-negative integer linear combination of the coordinates of $\boldsymbol{a}$. Marklof in [The asymptotic distribution of Frobenius numbers, Invent. Math. 181 (2010), 179–207] proved the existence of the limit distribution of the Frobenius numbers, when $\boldsymbol{a}$ is taken to be random in an enlarging domain in $\mathbb{R}^{d}$. We will show that if the domain has piecewise smooth boundary, the error term for the convergence of the distribution function is at most a polynomial in the enlarging factor.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathcal{O}$ be a maximal order in the quaternion algebra $B_p$ over $\mathbb{Q}$ ramified at $p$ and $\infty $. The paper is about the computational problem: construct a supersingular elliptic curve $E$ over $\mathbb{F}_p$ such that ${\rm End}(E) \cong \mathcal{O}$. We present an algorithm that solves this problem by taking gcds of the reductions modulo $p$ of Hilbert class polynomials.

New theoretical results are required to determine the complexity of our algorithm. Our main result is that, under certain conditions on a rank three sublattice $\mathcal{O}^T$ of $\mathcal{O}$, the order $\mathcal{O}$ is effectively characterized by the three successive minima and two other short vectors of $\mathcal{O}^T\! .$ The desired conditions turn out to hold whenever the $j$-invariant $j(E)$, of the elliptic curve with ${\rm End}(E) \cong \mathcal{O}$, lies in $\mathbb{F}_p$. We can then prove that our algorithm terminates with running time $O(p^{1+\varepsilon })$ under the aforementioned conditions.

As a further application we present an algorithm to simultaneously match all maximal order types with their associated $j$-invariants. Our algorithm has running time $O(p^{2.5 + \varepsilon })$ operations and is more efficient than Cerviño’s algorithm for the same problem.

We present a higher-dimensional generalization of the Gama–Nguyen algorithm (STOC ’08) for approximating the shortest vector problem in a lattice. This generalization approximates the densest sublattice by using a subroutine solving the exact problem in low dimension, such as the Dadush–Micciancio algorithm (SODA ’13). Our approximation factor corresponds to a natural inequality on Rankin’s constant derived from Rankin’s inequality.

In this paper, we present a heuristic algorithm for solving exact, as well as approximate, shortest vector and closest vector problems on lattices. The algorithm can be seen as a modified sieving algorithm for which the vectors of the intermediate sets lie in overlattices or translated cosets of overlattices. The key idea is hence no longer to work with a single lattice but to move the problems around in a tower of related lattices. We initiate the algorithm by sampling very short vectors in an overlattice of the original lattice that admits a quasi-orthonormal basis and hence an efficient enumeration of vectors of bounded norm. Taking sums of vectors in the sample, we construct short vectors in the next lattice. Finally, we obtain solution vector(s) in the initial lattice as a sum of vectors of an overlattice. The complexity analysis relies on the Gaussian heuristic. This heuristic is backed by experiments in low and high dimensions that closely reflect these estimates when solving hard lattice problems in the average case.

This new approach allows us to solve not only shortest vector problems, but also closest vector problems, in lattices of dimension $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$ in time $2^{0.3774\, n}$ using memory $2^{0.2925\, n}$. Moreover, the algorithm is straightforward to parallelize on most computer architectures.

We solve a randomized version of the following open question: is there a strictly convex, bounded curve $\gamma \subset { \mathbb{R} }^{2} $ such that the number of rational points on $\gamma $, with denominator $n$, approaches infinity with $n$? Although this natural problem appears to be out of reach using current methods, we consider a probabilistic analogue using a spatial Poisson process that simulates the refined rational lattice $(1/ d){ \mathbb{Z} }^{2} $, which we call ${M}_{d} $, for each natural number $d$. The main result here is that with probability $1$ there exists a strictly convex, bounded curve $\gamma $ such that $\vert \gamma \cap {M}_{d} \vert \rightarrow + \infty , $ as $d$ tends to infinity. The methods include the notion of a generalized affine length of a convex curve as defined by F. V. Petrov [Estimates for the number of rational points on convex curves and surfaces. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 344 (2007), 174–189; Engl. transl. J. Math. Sci. 147(6) (2007), 7218–7226].

Gama and Nguyen [‘Finding short lattice vectors within Mordell’s inequality’, in: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, New York, 2008, 257–278] have presented slide reduction which is currently the best SVP approximation algorithm in theory. In this paper, we prove the upper and lower bounds for the ratios $\Vert { \mathbf{b} }_{i}^{\ast } \Vert / {\lambda }_{i} (\mathbf{L} )$ and $\Vert {\mathbf{b} }_{i} \Vert / {\lambda }_{i} (\mathbf{L} )$, where ${\mathbf{b} }_{1} , \ldots , {\mathbf{b} }_{n} $ is a slide reduced basis and ${\lambda }_{1} (\mathbf{L} ), \ldots , {\lambda }_{n} (\mathbf{L} )$ denote the successive minima of the lattice $\mathbf{L} $. We define generalised slide reduction and use slide reduction to approximate $i$-SIVP, SMP and CVP. We also present a critical slide reduced basis for blocksize 2.

Let N(ρ; ω) be the number of points of a d-dimensional lattice Γ. where d≥2, inside a ball of radius ρ centred at the point ω. Denote by (ρ) the number N(ρ; ω) averaged over ω in the elementary cell Ω of the lattice Γ. The main result is the following lower bound for for dimensions d ≅ l(mod 4):

Existence criteria are presented for non-linear boundary value problems on the half line. In particular, the theory includes a problem in the theory of colloids and a problem arising in the unsteady flow of a gas through a semi-infinite porous medium.