This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published account of the now completed Flyspeck project.

Erdős asked the following question: given n points in the plane in almost general position (no four collinear), how large a set can we guarantee to find that is in general position (no three collinear)? Füredi constructed a set of n points in almost general position with no more than o(n) points in general position. Cardinal, Tóth and Wood extended this result to ℝ3, finding sets of n points with no five in a plane whose subsets with no four points in a plane have size o(n), and asked the question for higher dimensions: for given n, is it still true that the largest subset in general position we can guarantee to find has size o(n)? We answer their question for all d and derive improved bounds for certain dimensions.

Answering a question of Füredi and Loeb [On the best constant for the Besicovitch covering theorem. Proc. Amer. Math. Soc.121(4) (1994), 1063–1073], we show that the maximum number of pairwise intersecting homothets of a $d$-dimensional centrally symmetric convex body $K$, none of which contains the center of another in its interior, is at most $O(3^{d}d\log d)$. If $K$ is not necessarily centrally symmetric and the role of its center is played by its centroid, then the above bound can be replaced by $O(3^{d}\binom{2d}{d}d\log d)$. We establish analogous results for the case where the center is defined as an arbitrary point in the interior of $K$. We also show that, in the latter case, one can always find families of at least $\unicode[STIX]{x1D6FA}((2/\sqrt{3})^{d})$ translates of $K$ with the above property.

We consider finite point subsets (distributions) in compact metric spaces. In the case of general rectifiable metric spaces, non-trivial bounds for sums of distances between points of distributions and for discrepancies of distributions in metric balls are given (Theorem 1.1). We generalize Stolarsky’s invariance principle to distance-invariant spaces (Theorem 2.1). For arbitrary metric spaces, we prove a probabilistic invariance principle (Theorem 3.1). Furthermore, we construct equal-measure partitions of general rectifiable compact metric spaces into parts of small average diameter (Theorem 4.1).

Let $n\geqslant C$ for a large universal constant $C>0$ and let $B$ be a convex body in $\mathbb{R}^{n}$ such that for any $(x_{1},x_{2},\ldots ,x_{n})\in B$, any choice of signs $\unicode[STIX]{x1D700}_{1},\unicode[STIX]{x1D700}_{2},\ldots ,\unicode[STIX]{x1D700}_{n}\in \{-1,1\}$ and for any permutation $\unicode[STIX]{x1D70E}$ on $n$ elements, we have $(\unicode[STIX]{x1D700}_{1}x_{\unicode[STIX]{x1D70E}(1)},\unicode[STIX]{x1D700}_{2}x_{\unicode[STIX]{x1D70E}(2)},\ldots ,\unicode[STIX]{x1D700}_{n}x_{\unicode[STIX]{x1D70E}(n)})\in B$. We show that if $B$ is not a cube, then $B$ can be illuminated by strictly less than $2^{n}$ sources of light. This confirms the Hadwiger–Gohberg–Markus illumination conjecture for unit balls of $1$-symmetric norms in $\mathbb{R}^{n}$ for all sufficiently large $n$.

A lattice walk with all steps having the same length $d$ is called a $d$-walk. Denote by ${\mathcal{T}}_{d}$ the terminal set, that is, the set of all lattice points that can be reached from the origin by means of a $d$-walk. We examine some geometric and algebraic properties of the terminal set. After observing that $({\mathcal{T}}_{d},+)$ is a normal subgroup of the group $(\mathbb{Z}^{N},+)$, we ask questions about the quotient group $\mathbb{Z}^{N}/{\mathcal{T}}_{d}$ and give the number of elements of $\mathbb{Z}^{2}/{\mathcal{T}}_{d}$ in terms of $d$. To establish this result, we use several consequences of Fermat’s theorem about representations of prime numbers of the form $4k+1$ as the sum of two squares. One of the consequences is the fact, observed by Sierpiński, that every natural power of such a prime number has exactly one relatively prime representation. We provide explicit formulas for the relatively prime integers in this representation.

As an application of the boundary parametrization developed in our previous papers, we propose a new method to deduce information on the connected components of the interior of tiles. This gives a systematic way to study the topology of a certain class of self-affine tiles. An example due to Bandt and Gelbrich is examined to prove the efficiency of the method.

In this note we generalize a recent theorem of Guth and Katz on incidences between points and lines in 3-space from characteristic 0 to characteristic $p$ , and we explain how some of the special features of algebraic geometry in characteristic $p$ manifest themselves in problems of incidence geometry.

A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least (1−o(1))n 2.

We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.)

An important ingredient of our proofs is the following statement. Let S be a family of n open curves in ℝ2, so that each curve is the graph of a continuous real function defined on ℝ, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is $\Omega(nt\sqrt{\log t/\log\log t})$ .

We modify an argument of Hablicsek and Scherr to show that if a collection of points in $\mathbb{C}^{d}$ spans many $r$ -rich lines, then many of these lines must lie in a common $(d-1)$ -flat. This is closely related to a previous result of Dvir and Gopi.

We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75 (2006), 1449–1466]. By well-known decompositions, it is sufficient to consider the case of affine cones $s+\mathfrak{c}$ , where $s$ is an arbitrary real vertex and $\mathfrak{c}$ is a rational polyhedral cone. For a given rational subspace  $L$ , we define the intermediate generating functions $S^{L}(s+\mathfrak{c})(\unicode[STIX]{x1D709})$ by integrating an exponential function over all lattice slices of the affine cone  $s+\mathfrak{c}$ parallel to the subspace  $L$ and summing up the integrals. We expose the bidegree structure in parameters $s$ and $\unicode[STIX]{x1D709}$ , which was implicitly used in the algorithms in our papers [Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra. Found. Comput. Math.12 (2012), 435–469] and [Intermediate sums on polyhedra: computation and real Ehrhart theory. Mathematika59 (2013), 1–22]. The bidegree structure is key to a new proof for the Baldoni–Berline–Vergne approximation theorem for discrete generating functions [Local Euler–Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of rational polytopes. Contemp. Math.452 (2008), 15–33], using the Fourier analysis with respect to the parameter  $s$ and a continuity argument. Our study also enables a forthcoming paper, in which we study intermediate sums over multi-parameter families of polytopes.

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.

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.

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.

We show that if a collection of lines in a vector space over a finite field has “dimension” at least $2(d-1)+\unicode[STIX]{x1D6FD}$ , then its union has “dimension” at least $d+\unicode[STIX]{x1D6FD}$ . This is the sharp estimate of its type when no structural assumptions are placed on the collection of lines. We also consider some refinements and extensions of the main result, including estimates for unions of $k$ -planes.

Let p1, p2, p3 be three noncollinear points in the plane, and let P be a set of n other points in the plane. We show that the number of distinct distances between p1, p2, p3 and the points of P is Ω(n6/11), improving the lower bound Ω(n0.502) of Elekes and Szabó [4] (and considerably simplifying the analysis).

In this paper, we consider the so-called “Furstenberg set problem” in high dimensions. First, following Wolff’s work on the two-dimensional real case, we provide “reasonable” upper bounds for the problem for $\mathbb{R}$ or $\mathbb{F}_{p}$. Next we study the “critical” case and improve the “trivial” exponent by ${\rm\Omega}(1/n^{2})$ for $\mathbb{F}_{p}^{n}$. Our key tool in obtaining this lower bound is a theorem about how things behave when the Loomis–Whitney inequality is nearly sharp, as it helps us to reduce the problem to dimension two.

Given a finite-dimensional Banach space $X$ and an Auerbach basis $\{(x_{k},x_{k}^{\ast }):1\leqslant k\leqslant n\}$ of $X$, it is proved that there exist $n+1$ linear combinations $z_{1},\ldots ,z_{n+1}$ of $x_{1},\ldots ,x_{n}$ with coordinates $0,\pm 1$, such that $\Vert z_{k}\Vert =1$, for $k=1$, $2,\ldots ,n+1$ and $\Vert z_{k}-z_{l}\Vert >1$, for $1\leqslant k<l\leqslant n+1$.

We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve the bounds for the classical problem of packing identical spheres.

We investigate subgroups of $\text{SL}(n,\mathbb{Z})$ which preserve an open nondegenerate convex cone in $\mathbb{R}^{n}$ and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on self-dual cones, Weyl groups of certain Kac–Moody algebras, and they do occur in algebraic geometry as the automorphism groups of projective manifolds acting on their ample cones.