On generating functions in additive number theory, II: Lower-order terms and applications to PDEs
| Autor: | Scott T. Parsell, George Shakan, Julia Brandes, C. Poulias, Robert C. Vaughan |
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| Rok vydání: | 2020 |
| Předmět: |
11L15
11P55 35Q53 35Q55 Mathematics - Number Theory General Mathematics 010102 general mathematics Lower order 01 natural sciences Article Combinatorics 35Q55 Mathematics - Analysis of PDEs 35Q53 0103 physical sciences Additive number theory FOS: Mathematics 010307 mathematical physics Number Theory (math.NT) 0101 mathematics Mathematics 11L15 11P55 Analysis of PDEs (math.AP) |
| Zdroj: | Mathematische Annalen |
| DOI: | 10.48550/arxiv.2001.05629 |
| Popis: | We obtain asymptotics for sums of the form $$\begin{aligned} \sum _{n=1}^P e\left( {\alpha }_k\,n^k\,+\,{\alpha }_1 n\right) , \end{aligned}$$ ∑ n = 1 P e α k n k + α 1 n , involving lower order main terms. As an application, we show that for almost all $${\alpha }_2 \in [0,1)$$ α 2 ∈ [ 0 , 1 ) one has $$\begin{aligned} \sup _{{\alpha }_{1} \in [0,1)} \Big | \sum _{1 \le n \le P} e\left( {\alpha }_{1}\left( n^{3}+n\right) + {\alpha }_{2} n^{3}\right) \Big | \ll P^{3/4 + \varepsilon }, \end{aligned}$$ sup α 1 ∈ [ 0 , 1 ) | ∑ 1 ≤ n ≤ P e α 1 n 3 + n + α 2 n 3 | ≪ P 3 / 4 + ε , and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrödinger and Airy equations. |
| Databáze: | OpenAIRE |
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