Exponential Decay of Intersection Volume With Applications on List-Decodability and Gilbert-Varshamov Type Bound

We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking, this happens when the is (i) expanding and (ii) well-spread, (iii) certain random variable on boundary of ball has tail. As applications, we show that volume intersection Hamming, Johnson spaces symmetric groups decay exponentially as their centers drift apart. To verify condition (iii), prove large deviation inequalities ‘on slice’ functions with Lipschitz conditions. then use these estimates volumes 1) obtain sharp lower bound list-decodability q-ary codes, confirming conjecture Li Wootters, 2) improve classical Levenshtein from 1971 constant weight codes by factor linear dimension, resolving problem raised Jiang Vardy. Our probabilistic point view also offers unified framework improvements other Gilbert-Varshamov type bounds, giving conceptually simple calculation-free proofs $q$ -ary permutation spherical codes. Another consequence counting result number showing ampleness