Asymptotically sharp tradeoff for approximately dominating sets

Determine the asymptotically sharp relationship between the margin parameter [?] and the minimum committee size for approximately dominating sets; in particular, prove or refute that the upper bound k_{1/2-[?]}=O(1/[?]^2) is asymptotically correct, given the established lower bound [?](1/[?]) and upper bound O(1/[?]^2).

Background

The paper defines k_[?] as the smallest integer such that every election has an [?]-dominating set of size at most k_[?]. Its main theorem establishes that k_{1/2-[?]} is O(1/[?]2), while the lower-bound construction in Appendix A establishes k_{1/2-[?]}=?. Thus, the precise asymptotic dependence on the distance [?] from the Condorcet-dominating threshold 1/2 remains unresolved.

References

While our results show that $k_\alpha$ is finite for all $\alpha < \frac12$, it would be interesting to precisely pin down the asymptotic tradeoff between $\alpha$ and $k_\alpha$. In particular, our best bounds are $ \Omega(\frac{1}{\eps}) \leq k_{\frac12 - \eps} \leq O(\frac{1}{\eps2})$ by \Cref{thm:near-dom-sets,thm:lb}. We conjecture that the upper bound is correct.

— Approximately Dominating Sets in Elections  (2504.20372 - Charikar et al., 29 Apr 2025) in Section 6, Discussion, paragraph 'Sharper asymptotic bounds'