Threshold for the σ^−k regime of DRσ(k,w) (third range)

Prove the conjecture that the onset of the regime in which the expected density satisfies DRσ(k,w) ≤ C·σ^−k for some absolute constant C>1 occurs when w grows at least on the order of Ω(σ^k·k ln σ), i.e., establish that w ≥ c·σ^k·k ln σ (for some constant c>0) suffices to enter this “third range.”

Background

To describe the behavior of DRσ(k,w) across window sizes, the authors discuss three ranges: (i) DRσ(k,w) ≈ (2+o(1))/w, (ii) DRσ(k,w) ≤ C/w for absolute C>2, and (iii) DRσ(k,w) ≤ C/σ^k for absolute C>1.

They cite prior bounds for ranges (i) and (ii) and then state a conjecture for range (iii), proposing that w should be at least on the order of σ^k·k ln σ to achieve DRσ(k,w) ≤ C/σ^k. This formulates a concrete threshold conjecture for the σ^−k regime.