Lattice Points on Circles, Squares in Arithmetic Progressions and Sumsets of Squares

Let σ(k) denote the maximum of the number of squares in a+b, . . . , a+kb as we vary over positive integers a and b. Erdős conjectured that σ(k) = o(k) which Szemerédi [30] elegantly proved as follows: If there are more than δk squares amongst the integers a+b, . . . , a+kb (where k is sufficiently large) then there exists four indices 1 ≤ i1 < i2 < i3 < i4 ≤ k in arithmetic progression such that each a+ ijb is a square, by Szemerédi’s theorem. But then the a+ ijb are four squares in arithmetic progression, contradicting a result of Fermat. This result can be extended to any given field L which is a finite extension of the rational numbers: From Faltings’ theorem we know that there are only finitely many six term arithmetic progressions of squares in L, so from Szemerédi’s theorem we again deduce that there are oL(k) squares of elements of L in any k term arithmetic progression of numbers in L. (Xavier Xarles [31] recently proved that are never six squares in arithmetic progression in Z[ √ d] for any d.)