Unipotent Flows and Applications

We should think of the coefficients aij of Q as real numbers (not necessarily rational or integer). One can still ask what will happen if one substitutes integers for the xi. It is easy to see that if Q is a multiple of a form with rational coefficients, then the set of values Q(Z) is a discrete subset of R. Much deeper is the following conjecture: Conjecture 1.1 (Oppenheim, 1929). Suppose Q is not proportional to a rational form and n ≥ 5. Then Q(Z) is dense in the real line. This conjecture was extended by Davenport to n ≥ 3. Theorem 1.2 (Margulis, 1986). The Oppenheim Conjecture is true as long as n ≥ 3. Thus, if n ≥ 3 and Q is not proportional to a rational form, then Q(Z) is dense in R. This theorem is a triumph of ergodic theory. Before Margulis, the Oppenheim Conjecture was attacked by analytic number theory methods. (In particular it was known for n ≥ 21, and for diagonal forms with n ≥ 5). Failure of the Oppenheim Conjecture in dimension 2. Let α > 0 be a quadratic irrational such that α ̸∈ Q (e.g. α = (1 + √ 5)/2), and let Q(x1, x2) = x 2 1 − αx2.