I want to say I am working on this problem in my free time and it won't affect my daily tasks. Is the following correct?
I am working on this problem in the background.
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in the background
2 behind the scenes, unnoticed, out of the limelight
Rosemary likes to stay in the background.
In the background is closer in meaning to being unseen, hidden not "free time". So I am working on this problem in the background sounds like you are trying to hide something or keep it a secret.
I would stick to free time, spare time or time off.
- free time
Time that can be spent on one's own activities rather than work.
I love to play football in my free time.
- spare time
Your spare time is the time during which you do not have to work and you can do whatever you like
• In her spare time she read books on cooking.
• I spend a lot of my spare time watching videos.
- time off
time for rest or recreation away from your usual work or studies
A more informal phrase is "on my down time".
Another phrase to consider is on the side:
on the side
1. in addition to your regular job or activities He drives a bus, but he's a tour guide on the side.
On the side suggests you have a primary task or problem that you're working on and that this problem is secondary.