With any luck, by the time their grandchildren are pensioners...

[Nonverbis]

Upstream Proficiency by Virginia Evans and Jenny Dooley

With any luck, by the time their grandchildren are pensioners that figure may have reached the magic 100.

Could you help me understand why 'any' is used here. I'd say 'some'. It is a positive declarative sentence. As far as I remember, 'any' is used in negative and interrogative sentences.

 

[emsr2d2]

You can treat "With any luck" as a fixed expression. We use it a lot in the UK. It means the same as "With luck".

 

[jutfrank]

As far as I remember, 'any' is used in negative and interrogative sentences.

Yes, but by no means exclusively.

As emsr2d2 says, treat with any luck as a fixed expression. It means something loosely like 'if things go the way I hope'.