The additive-square-free spectrum below 139
Abstract
For a finite real alphabet T, let g(T) be the supremum of the lengths of words over T containing no two adjacent nonempty blocks of equal length and equal sum. We classify all four-element alphabets with g(T)<139. Up to affine equivalence, these are the known endpoint-sum family and seven isolated classes. Exhaustive finite searches give the isolated values, while 77 explicit words prove that every remaining real alphabet admits a word of length 139. The latter argument uses exact intersections of finite unions of lines, so it covers irrational parameters as well. The cutoff is sharp because g({0,2,3,7})=139. We also obtain a sharp cutoff of 167 within the family {0,1,x,x+2}, x>1. The required supplement prints the word certificates and specifies the complete-search evidence.
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BibTeX
@misc{zhang2026additivesquarefree,
title = {{The additive-square-free spectrum below 139}},
author = {Enkai Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02215},
eprint = {2610.02215},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02215},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The additive-square-free spectrum below 139
AU - Enkai Zhang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02215
UR - https://arxiv.org/abs/2610.02215
AB - For a finite real alphabet T, let g(T) be the supremum of the lengths of words over T containing no two adjacent nonempty blocks of equal length and equal sum. We classify all four-element alphabets with g(T)<139. Up to affine equivalence, these are the known endpoint-sum family and seven isolated classes. Exhaustive finite searches give the isolated values, while 77 explicit words prove that every remaining real alphabet admits a word of length 139. The latter argument uses exact intersections of finite unions of lines, so it covers irrational parameters as well. The cutoff is sharp because g({0,2,3,7})=139. We also obtain a sharp cutoff of 167 within the family {0,1,x,x+2}, x>1. The required supplement prints the word certificates and specifies the complete-search evidence.
ER -