Ditto the above. And if there's a label on the tire, install it located at the valve to further aid in locating tire problems.
What size tire? Rated pressure vs inflated pressure? Double wall or single wall rim?
The "gasket" you mention is called a "rim strip" and it's important to get the right type, size, and correct installation.
If you're just picking up a lot of road debris you might try a different tire with more flat protection built in. Schwalbe and Continental make some good ones.
Some cyclists use tire liners, like Tuffy Liners.
It's not unusual to get many more flats on the rear than on the front.