拋物面上的Falconer內積問題
Other Title
Falconer’s problem for dot product on paraboloids
Journal
2025臺大學士論文獎
Date Issued
2025
Author(s)
Chun-Kai Tseng
Advisor
Abstract
We establish dimensional thresholds for dot product sets associated with compact subsets of translated paraboloids. Specifically,we prove that when the dimension of such a subset exceeds 5/4 = 3/2 - 1/4 in R3, and d/2 - 1/4 - 1/8d-4 in Rd for d≥4, its dot product set has positive Lebesgue measure.
This result demonstrates that if a compact set in Rd exhibits aparaboloidal structure, then the usual dimensional barrierof d/2 for dot product sets can be lowered for d≥3. Our work serves as the continuous counterpart of [1], which examines the finite field setting with partial reliance on the extension conjecture.
The key idea, closely following [1], is to reformulate the dot product set on the paraboloid as a variant of a distance set. This reformulation allows us to leverage state-of-the-art results from the pinned distance problem, as establishedin [6] for d = 2 and [2] for higher dimensions. Finally, we present explicit constructions and existence proofs that highlight the sharpness of our results.
Subjects
Falconer’sproblem
paraboloid,dotproductset
distanceset,harmonic analysis
geometricmeasuretheory
Publisher
國立臺灣大學數學系
Description
獎項：傅斯年獎；指導教授：沈俊嚴
Type
thesis
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