On the discrete part of the Dirichlet spectrum and a general isolation result

We will talk about an improvement of the theorem of Szekeres from Diophantine Approximation theory and a similar strengthened result for all the numbers at which the discrete part of the Dirichlet spectrum is reached.

In addition, we will discuss several statements about pairs of functions $f(x)> g(x)>0$ such that the existence of solutions$\frac{p}{q}$of Diophantine inequality$| \alpha -\frac{p}{q}|< \frac{f(q)}{q^2}$leads to the existence of solutions of inequality$| \alpha -\frac{p}{q}|<\frac{g(q)}{q^2}$.