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PreprintCC BY 4.0arXiv

Exact diameter compression for four-point sumset cardinalities

  • Enkai Zhang
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Abstract

The least interval diameter needed to realize every attainable cardinality of an h-fold sumset of a four-element integer set is binomh+22+(hbmod2) for 2≤ h≤10 and dh=(h−3)(h−1) for h≥11. We prove the formula uniformly for h≥17 and use complete finite searches, with a relation-rank coverage argument, for the remaining orders. For the lower bound, we show that every integer four-set with binomh+33−17 distinct h-fold sums has normalized diameter in {dh,dh+1,dh+2} when h≥9, and we construct sets attaining dh. For the upper bound, a relation-lattice index argument restricts larger minimum-diameter representatives to two or three active directions. Two general constructions and four special defects then provide short sets realizing every remaining cardinality.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026exact,
  title = {{Exact diameter compression for four-point sumset cardinalities}},
  author = {Enkai Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02216},
  eprint = {2610.02216},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02216},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Exact diameter compression for four-point sumset cardinalities
AU  - Enkai Zhang
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.02216
UR  - https://arxiv.org/abs/2610.02216
AB  - The least interval diameter needed to realize every attainable cardinality of an h-fold sumset of a four-element integer set is binomh+22+(hbmod2) for 2≤ h≤10 and d_h=(h−3)(h−1) for h≥11. We prove the formula uniformly for h≥17 and use complete finite searches, with a relation-rank coverage argument, for the remaining orders. For the lower bound, we show that every integer four-set with binomh+33−17 distinct h-fold sums has normalized diameter in {d_h,d_h+1,d_h+2} when h≥9, and we construct sets attaining d_h. For the upper bound, a relation-lattice index argument restricts larger minimum-diameter representatives to two or three active directions. Two general constructions and four special defects then provide short sets realizing every remaining cardinality.
ER  -