Minkowski sums with convex curves without pointwise Fourier decay
Abstract
Let Γ⊂ℝ2 be a compact convex graph and define T(Γ) = inf { t: dimH(E)>t Longrightarrow |E+Γ|>0 for every compact E⊂ℝ2 }. For a graph over an interval of positive length the smallest possible value is T(Γ)=1. We ask whether this optimal conclusion can hold when pointwise Fourier decay of arclength is unavailable. The answer is yes, even for strictly convex curves. We use the Fourier transform convention ν̂(ξ)=∫ e−2πi x·ξ dν(x). We construct a strictly convex Lipschitz graph Γ with T(Γ)=1 such that, for every nontrivial subarc Γ0 and every α>0, limsup|ξ|→∞ |ξ|α| H1|Γ0̂(ξ) |= ∞. We also give a convex example for which arclength on every nontrivial subarc fails even to be a Rajchman measure. The geometric mechanism behind these examples is a positive curved trace: if Γ contains a positive-length subset of a C2 curve whose curvature is bounded away from zero, then |E+Γ|>0 whenever dimH(E)>1. For a nondegenerate graph this gives T(Γ)=1. For convex graphs it implies, in particular, that T(Γ)=1 whenever the curvature measure has a nonzero absolutely continuous part. The positive-measure proofs are in physical space and use translated-tube intersections and elementary facts about convex functions. The same overlap estimates give Mattila-type lower bounds for the average lengths of the associated curve projections of neighborhoods under the positive curved-trace hypothesis. We also prove a dimension-one endpoint result for sets with a positive-length rectifiable part and formulate the main remaining question: whether every strictly convex Lipschitz graph has the optimal threshold T(Γ)=1.
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BibTeX
@misc{iosevich2026minkowski,
title = {{Minkowski sums with convex curves without pointwise Fourier decay}},
author = {Alex Iosevich and Zhangze Li and Eyvindur Palsson and Krystal Taylor and Alexia Yavicoli},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28770},
eprint = {2608.28770},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28770},
}
RIS
TY - UNPB
TI - Minkowski sums with convex curves without pointwise Fourier decay
AU - Alex Iosevich
AU - Zhangze Li
AU - Eyvindur Palsson
AU - Krystal Taylor
AU - Alexia Yavicoli
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28770
UR - https://arxiv.org/abs/2608.28770
AB - Let Γ⊂ℝ^2 be a compact convex graph and define T(Γ) = inf { t: dim_H(E)>t Longrightarrow |E+Γ|>0 for every compact E⊂ℝ^2 }. For a graph over an interval of positive length the smallest possible value is T(Γ)=1. We ask whether this optimal conclusion can hold when pointwise Fourier decay of arclength is unavailable. The answer is yes, even for strictly convex curves. We use the Fourier transform convention ν̂(ξ)=∫ e^−2πi x·ξ dν(x). We construct a strictly convex Lipschitz graph Γ with T(Γ)=1 such that, for every nontrivial subarc Γ_0 and every α>0, limsup_|ξ|→∞ |ξ|^α| H^1|_Γ_0̂(ξ) |= ∞. We also give a convex example for which arclength on every nontrivial subarc fails even to be a Rajchman measure. The geometric mechanism behind these examples is a positive curved trace: if Γ contains a positive-length subset of a C^2 curve whose curvature is bounded away from zero, then |E+Γ|>0 whenever dim_H(E)>1. For a nondegenerate graph this gives T(Γ)=1. For convex graphs it implies, in particular, that T(Γ)=1 whenever the curvature measure has a nonzero absolutely continuous part. The positive-measure proofs are in physical space and use translated-tube intersections and elementary facts about convex functions. The same overlap estimates give Mattila-type lower bounds for the average lengths of the associated curve projections of neighborhoods under the positive curved-trace hypothesis. We also prove a dimension-one endpoint result for sets with a positive-length rectifiable part and formulate the main remaining question: whether every strictly convex Lipschitz graph has the optimal threshold T(Γ)=1.
ER -