This may not be the right place to ask this but...

I notice that this always happens in similar places like the gym and the library I consistently go to. I'm pretty sure getting stared at happens to most people too.

Now I can say that I'm the least approachable, when I go out to places like this, especially the gym, I always like to just wear a cap and do my business. I just don't want to be bothered. But every time, I will always catch someone staring/looking at me and almost always this will be a woman. I caught a woman older than I am looking at me twice, being one with little to no experience and frankly one who has a mission to just do my own thing, how do I go about solving this?

Do I smile back? Approach her, wink my eye? What? I wear a cap, I assume no one actually sees me but I always get the stares- I'm lean, semi-regular, just a newb minding his own business but I get the stares but I always just look away and then look back. Any tips in possibly making something out of this?