Meaning of HOT ON SB'S / STH'S HEELS in English

HOT ON SB'S / STH'S HEELS

following sb/sth very closely :

He turned and fled with Peter hot on his heels.

Further successes came hot on the heels of her first best-selling novel.

Oxford Advanced Learner's English Dictionary.      Оксфордский английский словарь для изучающик язык на продвинутом уровне.