Diophantine inequalities with mixed powers,II

It is shown that if λ₁, …, λ₅ are non-zero real numbers, not all of the same sign, and at least one of the ratios $\text{λ}{}_{\mathrm{i}}\text{λ}{}_{\mathrm{j}}$ (1 ≤ j ≤ 3) is irrational then the values taken by $\text{λ}{}_1\text{x}{}_1{}^2\text{+ λ}{}_2\text{x}{}_2{}^2\text{+ λ}{}_3\text{x}{}_3{}^2\text{+ λ}{}_4\text{x}{}_4{}^3\text{+ λ}{}_5\text{x}{}_5{}^3$ for integer values of x₁, …, x₅ are everywhere dense on the real line. Similar results are proved for the polynomials $\text{λ}{}_1\text{x}{}_1{}^2\text{+ λ}{}_2\text{x}{}_1{}^2\text{+ λ}{}_3\text{x}{}_3{}^3\text{+ … + λ}{}_6\text{x}{}_6{}^3$ and $\text{λ}{}_1\text{x}{}_1{}^2\text{+ λ}{}_2\text{x}{}_2{}^2\text{+ λ}{}_3\text{x}{}_3{}^3\text{+ λ}{}_4\text{x}{}_4{}^3\text{+ λ}{}_5\text{x}{}_5{}^4\text{+ λ}{}_6\text{x}{}_6{}^4$.