On small values of indefinite diagonal quadratic forms at integer points in at least five variables

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

Values of isotropic quadratic forms at S-integral points

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Values of Quadratic Forms at Integral Points : an Elementary Approach

On common values of lacunary polynomials at integer points

For fixed ` ≥ 2, fixed positive integers m1 > m2 with gcd(m1,m2) = 1 and n1 > n2 > · · · > n` with gcd(n1, . . . , n`) = 1, and fixed rationals a1, a2, . . . , a`+1, b1, b2 which are all nonzero except for possibly a`+1, we show the finiteness of integral solutions x, y of the equation a1x n1 + · · ·+ a`x` + a`+1 = b1y + b2y , when n1 ≥ 3, m1 ≥ 2`(` − 1), and (n1, n2) 6= (m1,m2). In relation to...