Séance Séminaire

Séminaire Gaston Darboux

Values of binary quadratic forms and continued fraction expansion

It is well known that for a quadratic form Q in 3 or more variables, with real or complex coefficients, which is nondegenerate, indefinite and not a scalar multiple of a form with rational coefficients, the set of values of Q over points with integral coordinates (Gaussian integers in the complex case) is dense among all numbers (real or complex respectively). The analoguous statement is not true for binary quadratic forms, whether over reals or complex numbers. The question in this case can be related to continued fraction expansions, and the results on the theme will be discussed using the classical theory of continued fractions in the real case and some newly obtained results on continued fraction expansions in the case of complex numbers.