RECORD
Higher Order Oscillation and Uniform Distribution
Dynamical Systems · Submitted 26 Dec 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This follows the result we prove in this paper that for a fixed non-zero real number $\alpha$ and almost all real numbers $\beta >1$ (alternatively, for a fixed real number $\beta >1$ and almost all real numbers $\alpha$) and for all real polynomials $Q(x)$, sequences $\big(\alpha \beta^{n}g(\beta)+Q(n)\big)_{n\in \N}$ are uniformly distributed modulo $1$.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
It is known that the M\"obius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form $(e^{2\pi i \alpha \beta^{n}g(\beta)})_{n\in \N}$, for a non-decreasing twice differentiable function $g$ with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number $\alpha$ and almost all real numbers $\beta >1$ (alternatively, for a fixed real number $\beta >1$ and almost all real numbers $\alpha$) and for all real polynomials $Q(x)$, sequences $\big(\alpha \beta^{n}g(\beta)+Q(n)\big)_{n\in \N}$ are uniformly distributed modulo $1$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | It is known that the M\"obius function in number theory is higher order oscillating. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s1 | In this paper we show that there is another kind of higher order oscillating sequences in the form $(e^{2\pi i \alpha \beta^{n}g(\beta)})_{n\in \N}$, for a non-decreasing twice differentiable function $g$ with a mild condition. | contribution 0.76 | 0.94 | 0.30 | asserted 1.00 |
| s2 | This follows the result we prove in this paper that for a fixed non-zero real number $\alpha$ and almost all real numbers $\beta >1$ (alternatively, for a fixed real number $\beta >1$ and almost all real numbers $\alpha$) and for all real polynomials $Q(x)$, sequences $\big(\alpha \beta^{n}g(\beta)+Q(n)\big)_{n\in \N}$ are uniformly distributed modulo $1$. | result 0.99 | 0.92 | 0.77 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
It is known that the M\"obius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form $(e^{2\pi i \alpha \beta^{n}g(\beta)})_{n\in \N}$, for a non-decreasing twice differentiable function $g$ with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number $\alpha$ and almost all real numbers $\beta >1$ (alternatively, for a fixed real number $\beta >1$ and almost all real numbers $\alpha$) and for all real polynomials $Q(x)$, sequences $\big(\alpha \beta^{n}g(\beta)+Q(n)\big)_{n\in \N}$ are uniformly distributed modulo $1$.
Human review
How the literature uses it
3 arXiv papers cite this in 4 sentences
Papers citing it · 3 sentences without a confident label
Builds on it 1 paper
-
This extends a result of Akiyama and Jiang [ref] who proved that for almost every $β>1$, the sequence $(e^2πi β^n)_n∈$ is fully oscillating.
Construction of some Chowla sequences · Introduction · 1.00
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.