On uniform Diophantine exponents of lattices

After Dirichlet proved his fundamental theorem which gave birth to the theory of Diophantine approximation, the concept of a Diophantine exponent emerged. This concept naturally splits into two types - regular exponents and uniform ones. In the case of one real number the regular exponent is a very nice characteristic of a number's Diophantine properties, whereasthe uniform exponent happens to be degenerate, as for irrational numbers it attains only one single value. In our talk we will discuss Diophantine exponents of lattices, both regular and uniform ones. We will show that the uniform Diophantine exponent of a lattice, if properly defined, is nondegenerate and, moreover, leads to a modification of the uniform exponent of one real number, which turns out to be nontrivial.