Describe the new feature: I think the title is self-explanatory - the cursor (seen in the image below) should blink to the beat of the currently playing song.

Afaik caret loop duration is hardcoded on framework side (See https://github.com/ppy/osu-framework/blob/master/osu.Framework/Graphics/UserInterface/TextBox.cs#L380)
Doing this might not look great with high BPM tracks. Would need to try and see.
Doing this might not look great with high BPM tracks. Would need to try and see.
Could be scaled down similar to how CtB bananas are handled: keep dividing by 2 until you get to a reasonable number
I'm not sure how this would look so I'll suggest it:
If the current playing song is in a Kiai section, then make the text jump in time with the music. ( could be in time with the caret.)
By jump I mean, increase the thickness ( kinda like bold but not half as much )
I suggest making it togglable if it gets too annoying on any specific map/song.
Personally I'm quite opposed to messing with the text thickness, I'm 99% sure it'll be too distracting.
Doing this might not look great with high BPM tracks. Would need to try and see.
Could be scaled down similar to how CtB bananas are handled: keep dividing by 2 until you get to a reasonable number
What about maps with a high prime BPM? In those cases, it would probably be best to just revert to the standard blink-rate.
I think you all are overthinking this, as there already is an implementation of this behaviour in lazer, see https://github.com/ppy/osu/blob/master/osu.Game.Tests/Visual/UserInterface/TestSceneBeatSyncedContainer.cs. Pretty sure this can also be applied here.
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Personally I'm quite opposed to messing with the text thickness, I'm 99% sure it'll be too distracting.