Iterated-sumset spectra: The complete exponent law and its rank geometry
Abstract
For integers h,k≥ 1, let hA be the h-fold sumset of A and put ℛ(h,k)={|hA|:A⊂ℤ, |A|=k}. Previously, the fixed-cardinality exponent law was known only for k≤ 3; every fixed k≥ 4 remained open. We settle the problem in full by determining the complete fixed-cardinality exponent law: |ℛ(h,k)|=1,&k≤ 2, h,&k=3, hk−1+ok(1),&k≥ 4. Here ok(1)→ 0 as h→∞ with k fixed. More sharply, for fixed k≥ 4, an interval of length Θk(hk−1) contains at least hk−1−ok(1) attainable values. At k=4 we prove |ℛ(h,4)|=Θ(h3) with positive lower density in its ambient interval, disproving Nathanson's proposed o(h3) and O(h2) bounds. One bounded addition-table geometry drives these results, coupling Hilbert-energy amplification to optimal finite-observation compression. Every ordered real k-set (k≥ 2) has an integer model in [0,Ok(hk−2)] preserving every sum equality and strict comparison through degree h; the exponent k−2 is sharp. The universal label-realization length is therefore Θk(hk−2), one power sharper than Nathanson's Ok(hk−1) bound. For h≥ 2 and k≥ 3, minimum active rank equals realization-frequency codimension, exponent-shape codimension, and sampling-rarity exponent; a full-exponent family has maximal-rank witnesses with Cohen-Macaulay toric coordinate rings. At rank zero, for h≥ 2, it proves the conjectural OEIS A227589 formula binomh+22+1{2nmid h} for the least normalized diameter of a four-point Bh-set. It also gives exact fixed-(h,k) popularity laws for k-subsets of {1,…,q} as q→∞, resolving Nathanson's Problems 9 and 10.
Related
- Growth beyond exponent 3/2 for convexity and iterated sum sets
- Iterated Graph Systems (II): Bernoulli percolation and critical-exponent universality classes on hierarchical lattices
- Variable-Cliff Nielsen Geometry and an Exponent -4/3 Lower Bound for the Infinite-Cliff Diameter
- Collision Geometry of Joint Spectra
- Enumerative geometry of conics and double covers on projective complete intersections
BibTeX
@misc{shin2026iteratedsumset,
title = {{Iterated-sumset spectra: The complete exponent law and its rank geometry}},
author = {Henry Shin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01690},
eprint = {2609.01690},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01690},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Iterated-sumset spectra: The complete exponent law and its rank geometry
AU - Henry Shin
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01690
UR - https://arxiv.org/abs/2609.01690
AB - For integers h,k≥ 1, let hA be the h-fold sumset of A and put ℛ(h,k)={|hA|:A⊂ℤ, |A|=k}. Previously, the fixed-cardinality exponent law was known only for k≤ 3; every fixed k≥ 4 remained open. We settle the problem in full by determining the complete fixed-cardinality exponent law: |ℛ(h,k)|=1,&k≤ 2, h,&k=3, h^k−1+o_k(1),&k≥ 4. Here o_k(1)→ 0 as h→∞ with k fixed. More sharply, for fixed k≥ 4, an interval of length Θ_k(h^k−1) contains at least h^k−1−o_k(1) attainable values. At k=4 we prove |ℛ(h,4)|=Θ(h^3) with positive lower density in its ambient interval, disproving Nathanson's proposed o(h^3) and O(h^2) bounds. One bounded addition-table geometry drives these results, coupling Hilbert-energy amplification to optimal finite-observation compression. Every ordered real k-set (k≥ 2) has an integer model in [0,O_k(h^k−2)] preserving every sum equality and strict comparison through degree h; the exponent k−2 is sharp. The universal label-realization length is therefore Θ_k(h^k−2), one power sharper than Nathanson's O_k(h^k−1) bound. For h≥ 2 and k≥ 3, minimum active rank equals realization-frequency codimension, exponent-shape codimension, and sampling-rarity exponent; a full-exponent family has maximal-rank witnesses with Cohen-Macaulay toric coordinate rings. At rank zero, for h≥ 2, it proves the conjectural OEIS A227589 formula binomh+22+1_{2nmid h} for the least normalized diameter of a four-point B_h-set. It also gives exact fixed-(h,k) popularity laws for k-subsets of {1,…,q} as q→∞, resolving Nathanson's Problems 9 and 10.
ER -