There he received in 1963 his Ph.D. with Jonas Kubilius as primary advisor and Yuri Linnik as secondary advisor and with thesis entitled (in Russian) "Метрические теоремы о дыяфантавых приближение алгебраическими числами ограниченной степени" (Metric Theorems of Diophantine Approximations and Approximations by Algebraic Numbers of Bounded Degree).
Another important influence was the Leningrad number theorist Yuri Linnik, who was Sprindzuk's advisor for his Russian doctorate of sciences.
In 1969-71 he investigated the arithmetic properties of the Siegel hypergeometric E- functions with algebraic parameters and defined a wider class of E*-functions.
His detailed studies of the Thue equation in algebraic number fields proved to be useful for the effective solution of a wide class of Diophantine equations and allowed him to study the possibility of effective approximations to algebraic numbers both in archimedean and non-archimedean domains.
In 1970 he was an Invited Speaker at the ICM in Nice with talk New applications of analytic and p-adic methods in diophantine approximations.