Diophantine inequalities with mixed powers

It is shown that λ₁, λ₂,…, λ₆, μ are not all of the same sign and at least one ratio $\text{λ}{}_{\mathrm{i}}\text{λ}{}_{\mathrm{j}}$ is irrational then the values taken by $\text{λ}{}_1\text{x}{}_1{}^3\text{+ ? + λ}{}_6\text{x}{}_6{}^3\text{+ μy}{}^3$ for integer values of x₁ ,…, x₆, y are everywhere dense on the real line. A similar result holds for expressions of the form $\text{λ}{}_1\text{x}{}_1{}^3\text{+ ? + λ}{}_4\text{x}{}_4{}^3\text{+ μ}{}_1\text{y}{}_1{}^2\text{+ μ}{}_2\text{y}{}_2{}^3$.