Equal values of binary forms at integral points

Values of isotropic quadratic forms at S-integral points

© Foundation Compositio Mathematica, 1992, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier ...

Values of Quadratic Forms at Integral Points : an Elementary Approach

Values of Indefinite Quadratic Forms at Integral Points and Flows on Spaces of Lattices

This mostly expository paper centers on recently proved conjectures in two areas: A) A conjecture of A. Oppenheim on the values of real indefinite quadratic forms at integral points. B) Conjectures of Dani, Raghunathan, and Margulis on closures of orbits in spaces of lattices such as SLn(R)/SLn(Z) . At first sight, A) belongs to analytic number theory and B) belongs to ergodic and Lie theory, a...

Values of Binary Quadratic Forms at Integer Points and Schmidt Games

We prove that for any countable set A of real numbers, the set of binary indefinite quadratic forms Q such that the closure of Q ` Z2 ́ is disjoint from A has full Hausdorff dimension. Dedicated to S.G. Dani on the occasion of his 65th birthday

Powerfree Values of Binary Forms

Given a binary form f(x; y) 2 Z[x; y], we will be interested in nding the smallest k for which we can establish that there are in nitely many integers a and b such that f(a; b) is k free. Necessarily, we require the f has no xed kth prime power divisor. Until the nal section of this paper, we will also consider f to be irreducible. We set n = deg f . For k = 2, this problem has recently become ...