Kneserized Anticoncentration and Reverse Absorption for Graham's Rearrangement Conjecture

Abstract

We establish a Kneser-based anticoncentration estimate for uniform subset sums in composite cyclic groups. The estimate contains a periodic loss and is weaker than its prime-modulus counterpart. Nevertheless, together with known small- and large-set results, it proves that, for every fixed $t\geq2$ such that $\mathbb{Z}_t$ is strongly sequenceable and every sufficiently large prime $p$, every subset of $\mathbb{Z}_{tp}\setminus\{0\}$ has a valid ordering, thus establishing the analogue of Graham&#39;s rearrangement conjecture for this family of composite cyclic groups. We then identify the structural source of this loss. An inverse theorem shows that failure of the stabilizer-free growth underlying prime-type anticoncentration forces almost all of the set into a proper subgroup or one of its cosets. We exploit this structure by reverse absorption. Iterating the resulting dichotomy between non-periodic anticoncentration and structured concentration proves that every subset of \[ \mathbb{Z}_k\setminus\{0\}, \qquad k=\prod_{i=1}^{s}p_i^{e_i}, \qquad \sum_{i=1}^{s}e_i\leq L, \qquad p_1<\cdots<p_s\leqγp_1, \] admits a valid ordering whenever $L$ and $γ$ are fixed and the primes $p_i$ are sufficiently large.