bibkey: metya2026uniformity authors: Nilava Metya and Satyaki Mukherjee year: 2026 title: Approximate Uniformity in Finite Convolution Models doi: 10.48550/arXiv.2609.38243 url: https://arxiv.org/abs/2609.38243v1 claim: Conjecture 4.7 specifies the exact full L1 minimum 1/(2n-1) for all n>=3 over ordinary convolution of two independent real closed-simplex laws; equation (22) supplies an attaining family. strata_touched:

• D5/S3/TotalVariation/IndependentConvolutionL1Minimum license: CC-BY-SA-4.0 triage: anchor

Approximate uniformity of independent finite convolution

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The original v1 is arXiv:2609.38243v1, DOI 10.48550/arXiv.2609.38243.

Nilava Metya and Satyaki Mukherjee, Approximate Uniformity in Finite Convolution Models, arXiv:2609.38243v1, §1 and §4.3, define ordinary convolution of two independent probability vectors on a finite interval. Both factors range over the closed real simplex.

Equation (21) minimizes the full L1 distance from uniform on the ordinary output support. Conjecture 4.7 states its value is 1/(2n-1) for n≥3. Equation (22) gives an attaining family. The source leaves the matching lower bound conjectural.

The conjecture ranges over every n≥3 and every independent pair of laws in the closed real simplex on n inputs, including zero entries, irrational entries and asymmetric factors. Its ordinary convolution has all 2n−1 output indices, from 0 through 2n−2; the objective is the full L1 sum over all these outputs, including zero-probability outputs. Equation (22)’s attaining family belongs to this same source domain.

The source declares CC BY-SA 4.0. The source descriptions in this note are adapted and attributed under that license. The squared-L2, KL and Wasserstein results in the source concern different objectives and are not substituted for Conjecture 4.7.

The exact source-level resolution target and bounded literature scope are in preregistration #11735 and Problems/metya-mukherjee-independent-convolution-l1.md.