On a problem of Bochner and von Neumann

The following theorem is proved. If there exists an everywhere dense set $\Gamma$ such that every solution of the equation $v_t'=-iA^*v$ satisfying the condition $v(0)\in\Gamma$ is defined on the entire axis and is bounded, then every compact solution of the equation $u'_t=iAu$ is an almost-periodic function.