Arithmetic Progressions of Squares and Multiple Dirichlet Series

We study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. We show that this multiple Dirichlet series has meromorphic continuation to $\mathbb{C}^2$ and use Tauberian methods to obtain counts for arithmetic progressions of squares and rational points on $x^2+y^2=2$.Comment: 31 pages, (now incorporating helpful comments

Lattice points on circles, squares in arithmetic progressions and sumsets of squares

Rudin conjectured that there are never more than c N^(1/2) squares in an arithmetic progression of length N. Motivated by this surprisingly difficult problem we formulate more than twenty conjectures in harmonic analysis, analytic number theory, arithmetic geometry, discrete geometry and additive combinatorics (some old and some new) which each, if true, would shed light on Rudin's conjecture.Comment: 21 pages, preliminary version. Comments welcom

On symmetric square values of quadratic polynomials

We prove that there does not exist a non-square quadratic polynomial with integer coefficients and an axis of symmetry which takes square values for N consecutive integers for N=7 or N >= 9. At the opposite, if N <= 6 or N=8 there are infinitely many

Arithmetic progressions of four squares over quadratic fields

Let d be a squarefree integer. Does there exist four squares in arithmeti progression over Q( √ d)? We shall give a partial answer to this question, depending on the value of d. In the a rmative ase, we onstru t expli it arithmeti progressions onsisting of four squares over Q( √ d)Research of the first author was supported in part by grant MTM 2009-07291 (Ministerio de Educación y Ciencia, Spain) and CCG08- UAM/ESP–3906 (Universidad Autónoma de Madrid – Comunidad de Madrid, Spain). Research of the second author was supported in part by grant MTM 2006-01859 (Ministerio de Educación y Ciencia, Spain

Lattice points on circles, squares in arithmetic progressions and sumsets of squares

22 páginas, 1 figura.-- 2000 Mathematics Subject Classification:11N36.-- En: Andrew Granwille, Melvyn B. Nathanson y József Solymosi (Eds.).We discuss the relationship between various additive problems concerning squares.Peer reviewe