Tightness of interpolation-rate upper bounds for non-smooth ℓp tolerant testing

Prove that the interpolation-regime upper bounds for the minimax critical separation in tolerant testing under non-smooth ℓp norms—specifically, for (i) ℓp with p in (1,2) and (ii) ℓp with odd p>2 as stated in Lemma UpperBoundLpOddLessTwo and Lemma UpperBoundLpOddGtrTwo—are tight up to polylogarithmic factors. In other words, construct matching lower bounds (up to polylogarithmic terms) showing that the rates given there are minimax-optimal across the corresponding interpolation ranges of the tolerance parameter ε0.

Background

In the Gaussian sequence model, the paper establishes upper bounds for the minimax critical separation in tolerant testing under general ℓp norms. For non-smooth ℓp norms (p not an even integer), the authors derive three regimes—free tolerance, interpolation, and functional estimation—and provide upper bounds in each. They match the free-tolerance and functional-estimation regimes with lower bounds (up to polylogarithms), but for the interpolation regime, only upper bounds are currently available.

The authors explicitly conjecture that these interpolation-regime upper bounds are tight (minimax-optimal up to polylog factors). Establishing tightness would require proving matching lower bounds in the interpolation regime for p∈(1,2) and odd p>2, thereby completing the characterization of the tolerant testing rates for non-smooth ℓp norms.