Let be a real number $ a. $ For any real number $ p $ and natural number $ k, $ let be the set $$ A_k(p)=\{ px\in\mathbb{Z}\mid k=\lceil x \rceil \} . $$Find all real numbers $ b $ such that $ \# A_n(a)=\# A_n(b) , $ for any natural number $ n. $ $ \# $ denotes the cardinal. Eugen Păltânea