GA pilot reported an NMAC at GAI non-towered airport.

Narrative

On the ramp at GAI I listened to the ATIS; then contacted Potomac Clearance to get my IFR clearance. Winds were 330 at 6+ [kts.]; so while I was putting the plan in my GPS; I listened in on the common frequency; but didn't hear anyone. I was fully loaded and felt Runway 32 was optimal for me; and noticed one plane ahead of me earlier used Runway 32. I announced on the common frequency that I was taxiing to 32; and stopped at the hold short line. I contacted Clearance and said I was number one at Runway 32 at GAI; and they said to stand by. While I waited; I again listened; announced on CTAF that I was at Runway 32 but didn't hear anyone. When I got my clearance void time; I announced again on CTAF that I was departing Runway 32 and when I lifted off I finally saw the student pilot veering to his right (my left) about 500 ft. away and I turned opposite also in a steep climb. I never heard the guy; and obviously we didn't see one another till we were very close. He never said anything; and I chalked it up to a new pilot.Potomac Approach did ask me to call; which I did when we got to ZZZ; and I explained the situation. My take was that it was probably a new pilot who 'complained'; I gave my viewpoint; and thought we had cleared up the issue; until contacted a month and a half later. I'm not normally based at GAI; but have learned to be more vigilant; especially with all the flight training that goes on there. In the future; I'll probably wait longer after getting a void time; so to give every pilot the opportunity to hear and respond call outs.