Minimum modulus for the unique multiset-sum problem

Abstract

Fix $n \ge 2$. A set $A = \{a_0 < a_1 < \dots < a_{n-1}\}$ of $n$ residues in $\Z_N$ is \emph{valid mod $N$} if the all-ones multiset is the \emph{only} size-$n$ multiset drawn from $A$ whose sum is $p := \sum_i a_i \pmod N$. For the super-increasing set $A = \{2^k - 1 : 0 \le k \le n-1\}$ we determine the least valid modulus exactly: $\Nmin(n) = 2^{\,n} - 2^{\lfloor \log_2 n \rfloor}$ for all $n \ge 2$. Both directions of the proof are elementary, resting on a sharp minimal-digit-sum estimate for representations by binary coins, and the full theorem has been machine-checked in Lean~4/Mathlib for all $n$ (https://github.com/jarfo/min-modulus). We conjecture that no size-$n$ residue set admits a smaller valid modulus. This validity condition is exactly what makes the permanent of an $n \times n$ matrix equal to a single coefficient of a row-product polynomial modulo $x^N - 1$, extractable by a size-$N$ discrete Fourier (or number-theoretic) transform; the theorem thus identifies the smallest transform, $N \approx 2^n$, for which this evaluation is exact. That application -- and the resulting common framework for the classical formulas of Ryser and Glynn and this transform -- is developed in a companion paper [2].