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 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.
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