Some inequalities for nonhomogeneous quadratic forms

Abstract

For 0≤μ<1, functions f(μ) are obtained such that for any real indefinite quadratic form Q(x, y, z) of type (1,2) and determinant D and real x₀, y₀, z_o, the inequality μ(f(μ) D)⅓< Q(x + x₀, y + y₀, z + z₀)<(f((μ) D)⅓ has a solution in integers x, y, z. This result is used to prove that for any real quaternary from Q(x,y. z, t) of type (1,3) and determinant D and real numbers x₀, y₀, z₀, t₀, the inequality 0<Q(x + x₀, y + y₀, z + z₀, t + t₀)< 128/25(2√7-I) IDI)^¼ has a solution in integers x, y. z, and t.