Title: Systems of small linear forms and Diophantine approximation on manifolds,
Abstract: We develop the theory of Diophantine approximation for systems of
simultaneously small linear forms, which coefficients are drawn from any given
analytic non-degenerate manifolds. This setup originates from a problem of
Sprindžuk from the 1970s on approximations to several real numbers by
conjugate algebraic numbers. Our main result is a Khintchine type theorem,
which convergence case is established without usual monotonicity constrains and
the divergence case is proved for Hausdorff measures. The result encompasses
several previous findings and, within the setup considered, gives the best
possible improvement of a recent theorem of Aka, Breuillard, Rosenzweig and
Saxcé on extremality.