Fixed-Defect Inverse Theorems for Subset Sums

Abstract

Let A be an n-element set of positive real numbers, let FS(A) be its set of subset sums, and put T_n=binomn+12. For every fixed integer C≥−1 and all sufficiently large n, we classify the sets satisfying |FS(A)|≤ T_n+n+C+1. Each such set is commensurable. Its unique primitive integer normalisation B either satisfies ∑ B≤ T_n+n+C or belongs to an explicit exceptional family specified by a missing element m∈{1,2} and an integer partition of C+m or C+m+1. If P denotes the partition function, the exceptional family has exactly P(C+1)+2P(C+2)+P(C+3) primitive dilation classes. We also prove a local inverse theorem for bounded increment excess. If, for sufficiently large i, adjoining the largest element to the preceding i−1 elements creates only i+e new subset sums, where e is bounded, then the i-element set is a dilation of [1,i+e]_ℤ with exactly e elements deleted. Conversely, every such deletion pattern has increment excess e. The proof combines a stabiliser argument in ℝ/xℤ, Kneser's theorem, a quadratic subset-sum bound, and endpoint propagation. These arguments also give effective commensurability and a finite-state encoding. Together with earlier results for C≤−2, this completes the eventual fixed-defect classification for every integer C.

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BibTeX RIS

BibTeX

@misc{chen2026fixeddefect,
  title = {{Fixed-Defect Inverse Theorems for Subset Sums}},
  author = {Lizhong Chen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00044},
  eprint = {2609.00044},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00044},
}

RIS

TY  - UNPB
TI  - Fixed-Defect Inverse Theorems for Subset Sums
AU  - Lizhong Chen
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.00044
UR  - https://arxiv.org/abs/2609.00044
AB  - Let A be an n-element set of positive real numbers, let FS(A) be its set of subset sums, and put T_n=binomn+12. For every fixed integer C≥−1 and all sufficiently large n, we classify the sets satisfying |FS(A)|≤ T_n+n+C+1. Each such set is commensurable. Its unique primitive integer normalisation B either satisfies ∑ B≤ T_n+n+C or belongs to an explicit exceptional family specified by a missing element m∈{1,2} and an integer partition of C+m or C+m+1. If P denotes the partition function, the exceptional family has exactly P(C+1)+2P(C+2)+P(C+3) primitive dilation classes. We also prove a local inverse theorem for bounded increment excess. If, for sufficiently large i, adjoining the largest element to the preceding i−1 elements creates only i+e new subset sums, where e is bounded, then the i-element set is a dilation of [1,i+e]_ℤ with exactly e elements deleted. Conversely, every such deletion pattern has increment excess e. The proof combines a stabiliser argument in ℝ/xℤ, Kneser's theorem, a quadratic subset-sum bound, and endpoint propagation. These arguments also give effective commensurability and a finite-state encoding. Together with earlier results for C≤−2, this completes the eventual fixed-defect classification for every integer C.
ER  -

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