Abstract. We obtain sharp estimates for multidimensional generalisations of Vinogradov's mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed.

Abstract. We show that a non-singular integral form of degree d is soluble over the integers if and only if it is soluble over R and over Q p for all primes p, provided that the form has at least (d −

We show that any subset of [N ] of density at least (log log N ) −2 −157 contains a nontrivial progression of the form x, x + y, x + y 2 . This is the first quantitatively effective version of the Bergelson-Leibman polynomial Szemerédi theorem for a progression involving polynomials of differing degrees. In the course of the proof, we also develop a quantitative version of a special case of a concatenation theorem of Tao and Ziegler, with polynomial bounds.

We show that sets of integers lacking the configuration $x$, $x+y$, $x+y^2$ have at most polylogarithmic density.

We show that any subset of the squares of positive relative upper density contains non-trivial solutions to a translation-invariant linear equation in five or more variables, with explicit quantitative bounds. As a consequence, we establish the partition regularity of any diagonal quadric in five or more variables whose coefficients sum to zero. Unlike previous approaches, which are limited to equations in seven or more variables, we employ transference technology of Green to import bounds from the linear setting.

We obtain quantitative bounds in the polynomial Szemerédi theorem of Bergelson and Leibman, provided the polynomials are homogeneous and of the same degree. Such configurations include arithmetic progressions with common difference equal to a perfect kth power.

In this paper, we obtain quantitative estimates for the asymptotic density of subsets of the integer lattice Z 2 that contain only trivial solutions to an additive equation involving binary forms. In the process we develop an analogue of Vinogradov's mean value theorem applicable to binary forms.

We establish partition regularity of the generalised Pythagorean equation in five or more variables. Furthermore, we show how Rado's characterisation of a partition regular equation remains valid over the set of positive kth powers, provided the equation has at least (1 + o(1))k log k variables. We thus completely describe which diagonal forms are partition regular and which are not, given sufficiently many variables. In addition, we prove a supersaturated version of Rado's theorem for a linear equation restricted either to squares minus one or to logarithmically-smooth numbers.Part 2. Rado's criterion over squares and higher powers 9. The smooth homogeneous Bergelson-Leibman theorem 10. A supersaturated generalisation of both Roth and Sárközy's theorems 11. Pseudorandom Roth-Sárközy 12. The W -trick for smooth powers and a non-linear Roth-Sárközy theorem 13. Deducing partition regularity