Small aircraft pilot reported a near miss with another aircraft when entering the traffic pattern at a non-towered airport in VMC conditions. The pilot immediately turned left to avoid the other aircraft and then returned to enter the traffic pattern; only to encounter the other aircraft again. The reporter stated they did not hear any position calls from the other aircraft.

Narrative

I was returning from doing a practice ILS at ZZZ working Boston Approach to CQX. We were using traffic on our ForeFlight iPads. Boston was working the subject aircraft; Aircraft Y. This aircraft was behind us and descending going to CQX also. Aircraft Y reported us in sight and continued his descent. We got a traffic alert that the traffic was directly behind us on our heading and at our altitude and within a mile. I directed an immediate left-hand turn to avoid a collision. After he went by; we returned to our course to CQX; expecting to enter the traffic pattern for Runway 24 by crossing midfield 500 ft. above traffic pattern to a left downwind. We made our proper calls and as we crossed midfield; Aircraft Y was on the midfield downwind in our 12 o'clock position. Having not heard any position calls from him we were surprised. We continued at our altitude and only began to descend when we passed over him. We entered a wide downwind and flew a standard traffic pattern keeping him in sight. I was told by airport personnel that he was a frequent visitor and almost always causing problems in the traffic pattern.