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Item Open Access DIOPHANTINE APPROXIMATION WITH RESTRICTED NUMERATORS AND DENOMINATORS ON SEMISIMPLE GROUPS(arXivLabs, 2014) Gorodnik A.; Kadyrov Sh.We consider the problem of Diophantine approximation on semisimple algebraic groups by rational points with restricted numerators and denominators and establish a quantitative approximation result for all real points in the group by rational points with a prescribed denominator and an almost prime numerator.Item Open Access Diophantine approximation with restricted numerators and denominators on semisimple groups(Journal de Théorie des Nombres de Bordeaux 29, 2017) Kadyrov Sh.; Gorodnik A.We consider the problem of Diophantine approximation on semisimple algebraic groups by rational points with restricted numerators and denominators and establish a quantitative approximation result for all real points in the group by rational points with a prescribed denominator and an almost prime numerator.Item Open Access Algebraic numbers, hyperbolicity, and density modulo one(Journal of Number Theory, 2012) Gorodnik A.; Kadyrov S.We prove the density of the sets of the form λm 1 μn 1ξ1 +···+ λm k μn k ξk: m,n ∈ N modulo one, where λi and μi are multiplicatively independent algebraic numbers satisfying some additional assumptions. The proof is based on analysing dynamics of higher-rank actions on compact abelian groups.