A result of Simonovits and Sós states that for any fixed graph H and any ε > 0 there exists δ > 0 such that if G is an n-vertex graph with the property that every S ⊆ V(G) contains pe(H) |S|v(H) ± δ nv(H) labelled copies of H, then G is quasirandom in the sense that every S ⊆ V(G) contains $\frac{1}{2}$p|S|2± ε n2 edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on δ−1 which is a tower of twos of height polynomial in ε−1. We give an alternative proof of this theorem which avoids the regularity lemma and shows that δ may be taken to be linear in ε when H is a clique and polynomial in ε for general H. This answers a problem raised by Simonovits and Sós.

Given an L–space knot we show that its ϒ function is the Legendre transform of a counting function equivalent to the d–invariants of its large surgeries. The unknotting obstruction obtained for the ϒ function is, in the case of L–space knots, contained in the d–invariants of large surgeries. Generalisations apply for connected sums of L–space knots, which imply that the slice obstruction provided by ϒ on the subgroup of concordance generated by L–space knots is no finer than that provided by the d–invariants.

We solve the problem of giving sharp asymptotic bounds on the Hausdorff dimensions of certain sets of badly approximable matrices, thus improving results of Broderick and Kleinbock (preprint 2013) as well as Weil (preprint 2013), and generalising to higher dimensions those of Kurzweil ('51) and Hensley ('92). In addition we use our technique to compute the Hausdorff f-measure of the set of matrices which are not ψ-approximable, given a dimension function f and a function ψ : (0, ∞) → (0, ∞). This complements earlier work by Dickinson and Velani ('97) who found the Hausdorff f-measure of the set of matrices which are ψ-approximable.

We will determine the motivic cohomology H*,* (BSOn, $\mathbb{Z}$/2) with coefficients in $\mathbb{Z}$/2 of the classifying space of special orthogonal groups SOn over the complex numbers $\mathbb{C}$.

Tits has defined Kac–Moody groups for all root systems, over all commutative rings with unit. A central concept is the idea of a prenilpotent pair of (real) roots. In particular, writing down his group presentation explicitly would require knowing all the Weyl-group orbits of such pairs. We show that for the hyperbolic root system E10 there are so many orbits that any attempt at direct enumeration is impractical. Namely, the number of orbits of prenilpotent pairs having inner product k grows at least as fast as (constant) ⋅ k7 as k → ∞. Our purpose is to motivate alternate approaches to Tits' groups, such as the one in [2].

Consider weak approximation for 0-cycles on a smooth proper variety defined over a number field, it is conjectured to be controlled by the Brauer group of the variety. Let X be a Châtelet surface or a smooth compactification of a homogeneous space of a connected linear algebraic group with connected stabilizer. Let Y be a rationally connected variety. We prove that weak approximation for 0-cycles on the product X × Y is controlled by its Brauer group if it is the case for Y after every finite extension of the base field. We do not suppose the existence of 0-cycles of degree 1 neither on X nor on Y.

Taking the signature of the closure of a braid defines a map from the braid group to the integers. In 2005, Gambaudo and Ghys expressed the homomorphism defect of this map in terms of the Meyer cocycle and the Burau representation. In the present paper, we simultaneously extend this result in two directions, considering the multivariable signature of the closure of a coloured tangle. The corresponding defect is expressed in terms of the Maslov index and of the Lagrangian functor defined by Turaev and the first-named author.

We study the four-genus of linear combinations of torus knots: g4(aT(p, q) #-bT(p′, q′)). Fixing positive p, q, p′, and q′, our focus is on the behavior of the four-genus as a function of positive a and b. Three types of examples are presented: in the first, for all a and b the four-genus is completely determined by the Tristram–Levine signature function; for the second, the recently defined Upsilon function of Ozsváth–Stipsicz–Szabó determines the four-genus for all a and b; for the third, a surprising interplay between signatures and Upsilon appears.

Let K be a number field or a function field of characteristic 0, let φ ∈ K(z) with deg(φ) ⩾ 2, and let α ∈ ℙ1(K). Let S be a finite set of places of K containing all the archimedean ones and the primes where φ has bad reduction. After excluding all the natural counterexamples, we define a subset A(φ, α) of ℤ⩾0 × ℤ>0 and show that for all but finitely many (m, n) ∈ A(φ, α) there is a prime 𝔭 ∉ S such that ord𝔭(φm+n(α)−φm(α)) = 1 and α has portrait (m, n) under the action of φ modulo 𝔭. This latter condition implies ord𝔭(φu+v(α)−φu(α)) ⩽ 0 for (u, v) ∈ ℤ⩾0 × ℤ>0 satisfying u < m or v < n. Our proof assumes a conjecture of Vojta for ℙ1 × ℙ1 in the number field case and is unconditional in the function field case thanks to a deep theorem of Yamanoi. This paper extends earlier work of Ingram–Silverman, Faber–Granville and the authors.