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We prove a result on approximations to a real number θ by algebraic numbers of degree ≤ 2 in the case when we have certain information about the uniform Diophantine exponent ω̂ for the linear form x₀ + θx₁ + θ²x₂.

Let $\Theta =({\theta }_1,{\theta }_2,{\theta }_3)\in {ℝ}^3$. Suppose that $1,{\theta }_1,{\theta }_2,{\theta }_3$ are linearly independent over $\mathbb{Z}$. For Diophantine exponents $$\begin{array}{cc}\hfill \alpha \left(\Theta \right)& =sup\{\gamma >0:\underset{t\to +\infty }{lim\; sup}{t}^{\gamma }{\psi }_{\Theta }\left(t\right)<+\infty \},\hfill \\ \hfill \beta \left(\Theta \right)& =sup\{\gamma >0:\underset{t\to +\infty }{lim\; inf}{t}^{\gamma }{\psi }_{\Theta }\left(t\right)<+\infty \}\hfill \end{array}$$ we prove $$\beta \left(\Theta \right)\ge \frac{1}{2}\left(\frac{\alpha \left(\Theta \right)}{1-\alpha \left(\Theta \right)}+\sqrt{{\left(\frac{\alpha \left(\Theta \right)}{1-\alpha \left(\Theta \right)}\right)}^2+\frac{4\alpha \left(\Theta \right)}{1-\alpha \left(\Theta \right)}}\right)\alpha \left(\Theta \right).$$

We prove a result on the existence of linear forms of a given Diophantine type.