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sniffnoySo I was only recently made aware of this paper by Christopher Shriver. In it he takes Stefan Steinerberger's method, and shows that actually it can be used to get bounds on ||n||/(log n) that hold for almost all n... as long as you measure "almost all" using logarithmic density rather than natural density.
This means we need a name for the new type of approximate liminf's and limsup's that arise from this density! :) Perhaps liminfap*? limsupap*? limap*? :)
...of course the thing is that there are other densities beyond natural and logarithmic, so potentially one needs a more general notation. But those are like the most common, so, eh, I'm OK singling those two out. :)
This means we need a name for the new type of approximate liminf's and limsup's that arise from this density! :) Perhaps liminfap*? limsupap*? limap*? :)
...of course the thing is that there are other densities beyond natural and logarithmic, so potentially one needs a more general notation. But those are like the most common, so, eh, I'm OK singling those two out. :)
