Publications

How does the smallest additive basis for a set change when its elements must lie in ℚ, ℤ or ℕ? Restricting rational to integer bases can nearly double the minimum size, while requiring nonnegative integers can cause a logarithmic loss. Both effects are essentially sharp.

Ruzsa conjectured that, for finite A,B⊂ℤᵈ and any ε>0, there is an N=N(d,ε) such that if B is not covered by N parallel hyperplanes, then |A+B|^(1/d) ≥ |A|^(1/d) + (1−ε)|B|^(1/d). We prove this conjecture.

For equal-volume A and B and 0<t≤1/2, the threshold |tA+(1−t)B| < (1+tᵈ)|A| forces the convex hull of their union, after translation, to have bounded relative volume. The threshold 1+tᵈ is sharp; we also prove a counterpart for repeated sums.

A centrally symmetric convex progression in ℤᵈ can be enclosed in a generalised arithmetic progression at a cost of only d^{O(d)} in size. This is the discrete analogue of the size control in John’s theorem, with the correct order in the dimension.

We link Hadwiger’s covering problem to Bourgain’s slicing conjecture. Together with its later solution, our result gives a bound of the form (4−ε)ᵈ on the number of translates needed to cover a convex body by its interior, for some absolute ε>0 and sufficiently large d.

We derive graph-theoretic conditions under which a recognition network can exactly represent the posterior distribution of a generative Bayesian network, and identify obstructions to such exact inversion.

We settle planar Brunn–Minkowski stability with sharp square-root control. The key step compares the volume missing from the convex hull of A+B with the corresponding gaps for A and B, with a constant arbitrarily close to one.

If |A+A| ≤ (2ᵏ+δ)|A| for a finite A⊂ℤᵏ, then either A is covered by a bounded number of parallel hyperplanes or its smallest containing convex progression has at most Oₖ(δ)|A| points outside A.

We prove sharp L¹ inequalities for sup-convolution on convex domains in dimensions up to three. They give the correct constants in a linear Brunn–Minkowski stability estimate for hypographs, including a two-function version.

The extra volume added by averaging a set with itself controls, linearly, the volume missing from its convex hull. This resolves a conjecture of Figalli and Jerison on stability for homothetic regions.

In bridge-burning Cops and Robbers, an edge disappears whenever the robber crosses it. We determine the order of magnitude of the longest capture time for graphs needing at least three bridge-burning cops.

How large can the radius of a graph be when its minimum degree and shortest cycle length are fixed? We settle the exact triangle-free case, determine the right order for several larger girths, and relate the remaining even-girth problem to the Erdős girth conjecture.

In the eternal colouring game, players keep a random graph properly coloured through indefinitely many moves. We determine its typical game chromatic number for odd order and for even order when p=1/k, and prove an upper bound for other fixed edge probabilities.

Towers on the infinite square grid transmit a signal that weakens with distance. When the initial strength is t>17 and each point must receive a total strength of at least 3, we determine the smallest possible asymptotic density of towers, settling a conjecture of Drews, Harris and Randolph.

Tomaszewski conjectured that at least half of all signed sums of unit-normalised coefficients fall between −1 and 1. We improve the unconditional guarantee to 0.46.

We settle the complete-graph case of the long-standing bunkbed conjecture in bond percolation, when every edge has the same retention probability.

For n≥2, we construct uncountably many non-isomorphic partitions of ℤⁿ into 2n parts in which every point has exactly one neighbour in each part. This settles a question of Gross and Grupel.