Abstract. Given a family of varieties X → P n over a number field, we determine conditions under which there is a Brauer-Manin obstruction to weak approximation for 100% of the fibres which are everywhere locally soluble.

Harold Davenport was one of the truly great mathematicians of the twentieth century. Based on lectures he gave at the University of Michigan in the early 1960s, this book is concerned with the use of analytic methods in the study of integer solutions to Diophantine equations and Diophantine inequalities. It provides an excellent introduction to a timeless area of number theory that is still as widely researched today as it was when the book originally appeared. The three main themes of the book are Waring's problem and the representation of integers by diagonal forms, the solubility in integers of systems of forms in many variables, and the solubility in integers of diagonal inequalities. For the second edition of the book a comprehensive foreword has been added in which three prominent authorities describe the modern context and recent developments. A thorough bibliography has also been added.

Abstract. Let X be a projective non-singular quartic hypersurface of dimension 39 or more, which is defined over Q. We show that X(Q) is non-empty provided that X(R) is non-empty and X has p-adic points for every prime p.

We generalise Birch's seminal work on forms in many variables to handle a system of forms in which the degrees need not all be the same. This allows us to prove the Hasse principle, weak approximation, and the Manin-Peyre conjecture for a smooth and geometrically integral variety X ⊆ P m , provided only that its dimension is large enough in terms of its degree.

We revisit recent work of Heath-Brown on the average order of the quantity r(L 1 (x)) · · · r(L 4 (x)), for suitable binary linear forms L 1 , . . . , L 4 , as x = (x 1 , x 2 ) ranges over quite general regions in Z 2 . In addition to improving the error term in Heath-Brown's estimate, we generalise his result to cover a wider class of linear forms.