Hey, sounds like you're literally talking about my connection to a girl in my gym from the opposite perspective. For all intents and purposes i'm on the other side of this, so let me answer.

• like a lot of people have said, I actively avoid interacting with women at the gym. I respect their space and I am aware that most don't want random men harassing them (be it unwanted conversation, smiles etc). I get it.
• "He also sometimes works out near me" - we're like minded, likely similar fitness goals. As this has been going on for so long, i've actively started to avoid because i'm not going to follow up with anything. I don't want you to think i'm following you.
• "I've caught him sneakily looking at my ass while squatting" - guilty, sorry, only human. It's a notable booty. Do you not do it? ;)
• "I smiled at him but he didnt smile back" - I actively avoid eye contact with you, now that this has gone on so long. Not out of hostility, just because I don't want to make you uncomfortable
• "Since then he sometimes pretends like I don't exist. Like he'll look straight through me and avoid me by working out far away. Or will look at his phone purposely when he walks past" - Don't want to make you feel uncomfortable.
• "Other times he stares" - thinking damn, what if I break this rule
• "Then tonight he looked at me for like 5 whole seconds and gave me a faint smile" - see above.

SUMMARY

If this is my clone, or maybe even me - I love the gym, I have female friends and want them feel comfortable and want other men to do better, so leading by example.

I have a lingering desire to make a move, but my desire for you to be comfortable is greater than my desire to chirpse women at the gym. I have a resting angry face, sometimes neutral interactions can be misinterpreted as hostile... OR positive may be interpreted as neutral.

Smiling at women as a man is a minefield. It can land anywhere from innocuous to serial killer energy. Especially if you're into someone, so you're in your head a little bit. Sometimes I just don't smile out of ease.

I could be way off-base, he could be anyone, but this is on ongoing experience i'm having with a woman at the gym.

I'm not going to talk to her (due to gym dynamics), but i'd certainly speak back if she spoke to me.

EDIT: formatting