Gum
I guess if you're not swallowing it and you're not having issues with the sweeteners, I can't think of a reason to give up your gum. I can't chew gum because I always bite off little pieces and swallow it. Rather than have to keep vigilant, I just gave it up.
Chewing preps food for swallowing, so as you chew, your body prepares for food. When none comes, your body sends hunger signals for the food it was expecting, so chewing gum can trigger hunger. Some people aren't sensitive to this, but for others it's a real issue. To be safe, I was told to avoid it. During my pre-op phase, they asked how I was coping, and I told them I chewed gum to satisfy my urge to just chew on something when I wasn't really hungry. That's when they told me it's best to avoid it. I CAN have sugar free mints to help with the protein and ketosis breath.
Chewing gum stimulates the production of stomach acid...and that's the LAST thing I need since excess acid is a huge issue for me anyway. And excess acid in the tummy totally mimics hunger and makes me thing I'm starving. I'd also heard about the danger of it getting stuck.
I'd talk to your doctor about it. If you're not having acid/hunger issues and he's not afraid of you swallowing it, then it's probably not the worst thing you could do.
I was told not to chew it because it will put air into the stomach. I just use sugar free breath mints now, and do not chew them at all.