Small additive groups with A+ξA=ℝ
Abstract
We prove that, for every irrational ξ, any Fσ additive subgroup of R has an Fσ Q-vector space extension of the same Hausdorff dimension satisfying A+ξA=R. Iteration gives simultaneous surjectivity for any prescribed countable family of irrational multipliers. Such a space exists in every prescribed dimension d∈[0,1). We also construct in ZFC a zero-dimensional lightface Δ03 Q-vector subspace A for which A+ξA=R holds for every irrational ξ, answering both parts of Question~2 of Ye, Yu, and Zhao. This universal group is a difference of two Fσ sets, but is neither Fσ nor Gδ. Both constructions use direct inductions on finite binary strings and admit effective versions. As applications, we obtain zero-dimensional Fσ groups whose Cartesian squares have Hausdorff dimension one, and relate their dilate sums to Marstrand's projection theorem.
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BibTeX
@misc{yu2026small,
title = {{Small additive groups with $A+ξA=\mathbb R$}},
author = {Liang Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38970},
eprint = {2609.38970},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38970},
note = {CC0},
}
RIS
TY - UNPB
TI - Small additive groups with A+ξA=ℝ
AU - Liang Yu
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.38970
UR - https://arxiv.org/abs/2609.38970
AB - We prove that, for every irrational ξ, any F_σ additive subgroup of R has an F_σ Q-vector space extension of the same Hausdorff dimension satisfying A+ξA=R. Iteration gives simultaneous surjectivity for any prescribed countable family of irrational multipliers. Such a space exists in every prescribed dimension d∈[0,1). We also construct in ZFC a zero-dimensional lightface Δ^0_3 Q-vector subspace A for which A+ξA=R holds for every irrational ξ, answering both parts of Question~2 of Ye, Yu, and Zhao. This universal group is a difference of two F_σ sets, but is neither F_σ nor G_δ. Both constructions use direct inductions on finite binary strings and admit effective versions. As applications, we obtain zero-dimensional F_σ groups whose Cartesian squares have Hausdorff dimension one, and relate their dilate sums to Marstrand's projection theorem.
ER -