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I was trying to give people a sense of the statement, giving two examples of "not very dense" and "dense enough", but I don't understand what you're trying to say here. I can't work out whether you are asking a question, or making a conjecture, or ... what.

Would you care to clarify? If it's a question then I'll try to answer, but if it's a conjecture, can you make it more precise?

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He is asking "what is the line between too sparse and not too sparse" and attempting to characterize the example you gave of each in terms of approximate relative sparsity.

Right ... understood.

I don't have an answer, it's not my area, but it does feel a lot like the condition "sum(1/n) diverges" is likely to be close, but that's just a gut feeling.

Ah well there isn't one, from any divergent series you can construct a more slowly growing (but still divergent) series.

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