Subexponential factors and threshold behavior

Determine the subexponential factors in the number of direct discrete-Gaussian samples required for approximate and exact shortest-vector search on Haar-random unimodular lattices, and characterize the success behavior when the query exponent equals the threshold K_gamma^*.

Background

Theorem 1 establishes the sharp exponential sample threshold K_gamma^* for direct discrete-Gaussian search on Haar-random unimodular lattices, including adaptive choices of Gaussian widths and optional primitive reduction. It proves failure below the threshold and success above it, but does not resolve the finer behavior at the boundary or the multiplicative and other subexponential factors hidden by the exponential-rate formulation.

The unresolved issue concerns both the precise sample complexity beyond its leading exponential term and the critical regime in which the number of queries has exponent exactly K_gamma^*. These refinements would distinguish whether success converges to zero, one, or a nontrivial limiting value at equality and would quantify the finite-dimensional overhead of the direct-search mechanism.