Oppenheim conjecture

In Diophantine approximation, the Oppenheim conjecture concerns representations of numbers by real quadratic forms in several variables. It was formulated in 1929 by Alexander Oppenheim and later the conjectured property was further strengthened by Davenport and Oppenheim. Initial research on this problem took the number n of variables to be large, and applied a version of the Hardy-Littlewood circle method. The definitive work of Margulis, settling the conjecture in the affirmative, used methods arising from ergodic theory and the study of discrete subgroups of semisimple Lie groups.

Meyer's theorem states that an indefinite integral quadratic form Q in n variables, n ≥ 5, nontrivially represents zero, i.e. there exists a non-zero vector x with integer components such that Q(x) = 0. The Oppenheim conjecture can be viewed as an analogue of this statement for forms Q that are not multiples of a rational form. It states that in this case, the set of values of Q on integer vectors is a dense subset of the real line.

Several versions of the conjecture were formulated by Oppenheim and Davenport.

For n ≥ 5 this was conjectured by Oppenheim in 1929; the stronger version is due to Davenport in 1946.

This was conjectured by Oppenheim in 1953 and proved by Birch, Davenport, and Ridout for n at least 21, and by Davenport and Heilbronn for diagonal forms in five variables. Other partial results are due to Oppenheim (for forms in four variables, but under the strong restriction that the form represents zero over Z), Watson, Iwaniec, Baker–Schlickewey. Early work analytic number theory and reduction theory of quadratic forms.

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