C182 pilot experiences a NMAC with a light single on the left downwind to Runway 34 at BDN. TCAS equipment aboard the C182 showed the light single descending from 300 feet above and the light single pilot was so advised.

Narrative

Active runway at BDN at this time was 34. I made a 10-mile announcement from the northeast; then a 3-mile call that I was overflying the airfield east to west at 5;500 feet. My next call was overhead the airfield. As I got two miles west of the airfield; I made a right descending turn so that I was on a 45-degree approach to the left downwind leg of Runway 34. Prior to this point; there were no other aircraft announcing their intentions to enter the pattern at BDN. I announced my 45 degree approach to the left downwind to Runway 34 and immediately there was an announcement from a [light single] that he was 6 miles north and would enter the left downwind. As I entered and announced left downwind; he announced downwind; as well; and immediately showed up on my TCAS 300 feet right above me. I told him that I thought we were both in the same place and he executed a right 360 degree turn to re-enter the pattern behind me. We landed 1-2 with no other issues. The conditions were definitely VMC; however the area was experiencing forest fire smoke. I believe the [light single] was really northeast of the field when he made his first announcement; and was much closer than 6 miles. He also didn't enter the pattern using standard practice. Instead of overflying mid-field; he jogged to the right at the crosswind leg of Runway 34 and dove into the left downwind leg of the pattern. I made several position reports throughout this ordeal; and his first report was as he was entering the airport environment; just prior to downwind 34.