沈俊嚴Chun-Kai Tseng曾軍愷2025-11-142025-11-142025https://scholars.lib.ntu.edu.tw/handle/123456789/733686https://ntu.primo.exlibrisgroup.com/permalink/886NTU_INST/14poklj/alma991039401170104786獎項：傅斯年獎；指導教授：沈俊嚴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.我們研究了拋物面上的緊緻子集的內積集合，並對其維度條件進行分析。具體來說，我們證明：當該子集在 R3 中的 Hausdorff 維度超過 5/4 = 3/2 - 1/4，以及在 Rd 中（d≥4）超過 d/2 - 1/4 - 1/8d-4 時，其內積集合具有正的勒貝格測度。 這個結果顯示：若一個緊緻集合在 Rd 中具有拋物面結構，則對於 d≥3，內積集合的維度條件可以放寬，突破傳統上 d/2 的限制。我們的研究可以視為文獻 [1] 在有限域設定下結果的連續版本；該文部分依賴於一個未解的調和分析猜想。 我們的核心想法是，參考 [1] 的研究策略，將拋物面上的內積集合重新轉化為某種形式的距離集合。這樣的轉化讓我們能夠運用現有在 pinned distance 問題中的先進成果，包含 [6] 中針對 d = 2 的結果，以及 [2] 中處理高維度的研究。最後，我們也提出具體構造與存在性證明，以說明本結果在理論上的緊密程度。Falconer’sproblemparaboloid,dotproductsetdistanceset,harmonic analysisgeometricmeasuretheoryFalconer 問題拋物面內積集合距離集合調和分析幾何測度論thesis