Pub Date : 2023-01-01DOI: 10.4310/arkiv.2023.v61.n2.a8
Yuta Suzuki
Let $R_{k,ell}(N)$ be the representation function for the sum of the $k$-th power of a prime and the $ell$-th power of a positive integer. Languasco and Zaccagnini (2017) proved an asymptotic formula for the average of $R_{1,2}(N)$ over short intervals $(X,X+H]$ of the length $H$ slightly shorter than $X^{frac{1}{2}}$, which is shorter than the length $H=X^{frac{1}{2}+epsilon}$ in the exceptional set estimates of Mikawa (1993) and of Perelli and Pintz (1995). In this paper, we prove that the same asymptotic formula for $R_{1,2}(N)$ holds for $H$ of the size $X^{0.337}$. Recently, Languasco and Zaccagnini (2018) extended their result to more general $(k,ell)$. We also consider this general case, and as a corollary, we prove a conditional result of Languasco and Zaccagnini (2018) for the case $ell=2$ unconditionally up to some small factors.
{"title":"On the sum of a prime power and a power in short intervals","authors":"Yuta Suzuki","doi":"10.4310/arkiv.2023.v61.n2.a8","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n2.a8","url":null,"abstract":"Let $R_{k,ell}(N)$ be the representation function for the sum of the $k$-th power of a prime and the $ell$-th power of a positive integer. Languasco and Zaccagnini (2017) proved an asymptotic formula for the average of $R_{1,2}(N)$ over short intervals $(X,X+H]$ of the length $H$ slightly shorter than $X^{frac{1}{2}}$, which is shorter than the length $H=X^{frac{1}{2}+epsilon}$ in the exceptional set estimates of Mikawa (1993) and of Perelli and Pintz (1995). In this paper, we prove that the same asymptotic formula for $R_{1,2}(N)$ holds for $H$ of the size $X^{0.337}$. Recently, Languasco and Zaccagnini (2018) extended their result to more general $(k,ell)$. We also consider this general case, and as a corollary, we prove a conditional result of Languasco and Zaccagnini (2018) for the case $ell=2$ unconditionally up to some small factors.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135783630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/arkiv.2023.v61.n2.a1
Julia Brandes, Igor E. Shparlinski
We show that a certain two-dimensional family of Weyl sums of length $P$ takes values as large as $P^{3/4 + o(1)}$ on almost all linear slices of the unit torus, contradicting a widely held expectation that Weyl sums should exhibit square-root cancellation on generic subvarieties of the unit torus. This is an extension of a result of J. Brandes, S. T. Parsell, C. Poulias, G. Shakan and R. C. Vaughan (2020) from quadratic and cubic monomials to general polynomials of arbitrary degree. The new ingredients of our approach are the classical results of E. Bombieri (1966) on exponential sums along a curve and R. J. Duffin and A. C. Schaeffer (1941) on Diophantine approximations by rational numbers with prime denominators.
我们证明了长度为$P$的二维Weyl和族在几乎所有单位环面线性片上的值都高达$P^{3/4 + o(1)}$，这与人们普遍认为Weyl和在单位环面的一般子变异上应该表现平方根消去的期望相矛盾。这是J. Brandes, S. T. Parsell, C. Poulias, G. Shakan和R. C. Vaughan(2020)从二次多项式和三次多项式到任意次一般多项式的结果的推广。我们方法的新成分是E. Bombieri(1966)关于曲线上的指数和的经典结果，以及R. J. Duffin和a . C. Schaeffer(1941)关于以素数为分母的有理数的Diophantine近似的经典结果。
{"title":"Two-dimensional Weyl sums failing square-root cancellation along lines","authors":"Julia Brandes, Igor E. Shparlinski","doi":"10.4310/arkiv.2023.v61.n2.a1","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n2.a1","url":null,"abstract":"We show that a certain two-dimensional family of Weyl sums of length $P$ takes values as large as $P^{3/4 + o(1)}$ on almost all linear slices of the unit torus, contradicting a widely held expectation that Weyl sums should exhibit square-root cancellation on generic subvarieties of the unit torus. This is an extension of a result of J. Brandes, S. T. Parsell, C. Poulias, G. Shakan and R. C. Vaughan (2020) from quadratic and cubic monomials to general polynomials of arbitrary degree. The new ingredients of our approach are the classical results of E. Bombieri (1966) on exponential sums along a curve and R. J. Duffin and A. C. Schaeffer (1941) on Diophantine approximations by rational numbers with prime denominators.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/arkiv.2023.v61.n2.a9
Sailun Zhan
G"ottsche and Soergel gave formulas for the Hodge numbers of Hilbert schemes of points on a smooth algebraic surface and the Hodge numbers of generalized Kummer varieties. When a smooth projective surface $S$ admits an action by a finite group $G$, we describe the action of $G$ on the Hodge pieces via point counting. Each element of $G$ gives a trace on $sum_{n=0}^{infty}sum_{i=0}^{infty}(-1)^{i}H^{i}(S^{[n]},mathbb{C})q^{n}$. In the case that $S$ is a K3 surface or an abelian surface, the resulting generating functions give some interesting modular forms when $G$ acts faithfully and symplectically on $S$.
{"title":"Hilbert schemes of points on smooth projective surfaces and generalized Kummer varieties with finite group actions","authors":"Sailun Zhan","doi":"10.4310/arkiv.2023.v61.n2.a9","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n2.a9","url":null,"abstract":"G\"ottsche and Soergel gave formulas for the Hodge numbers of Hilbert schemes of points on a smooth algebraic surface and the Hodge numbers of generalized Kummer varieties. When a smooth projective surface $S$ admits an action by a finite group $G$, we describe the action of $G$ on the Hodge pieces via point counting. Each element of $G$ gives a trace on $sum_{n=0}^{infty}sum_{i=0}^{infty}(-1)^{i}H^{i}(S^{[n]},mathbb{C})q^{n}$. In the case that $S$ is a K3 surface or an abelian surface, the resulting generating functions give some interesting modular forms when $G$ acts faithfully and symplectically on $S$.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135659923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/arkiv.2023.v61.n2.a6
Omar Leon Sanchez, Susan J. Sierra
We consider when the symmetric algebra of an infinite-dimensional Lie algebra, equipped with the natural Poisson bracket, satisfies the ascending chain condition (ACC) on Poisson ideals. We define a combinatorial condition on a graded Lie algebra which we call Dicksonian because it is related to Dickson's lemma on finite subsets of $mathbb N^k$. Our main result is:
{"title":"A Poisson basis theorem for symmetric algebras of infinite-dimensional Lie algebras","authors":"Omar Leon Sanchez, Susan J. Sierra","doi":"10.4310/arkiv.2023.v61.n2.a6","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n2.a6","url":null,"abstract":"We consider when the symmetric algebra of an infinite-dimensional Lie algebra, equipped with the natural Poisson bracket, satisfies the ascending chain condition (ACC) on Poisson ideals. We define a combinatorial condition on a graded Lie algebra which we call Dicksonian because it is related to Dickson's lemma on finite subsets of $mathbb N^k$. Our main result is: ","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/arkiv.2023.v61.n2.a5
Kunyu Guo, Xianfeng Zhao, Dechao Zheng
In this paper, it is shown that some new phenomenon related to the spectra of Toeplitz operators with bounded harmonic symbols on the Bergman space. On one hand, we prove that the spectrum of the Toeplitz operator with symbol ${bar{z}+p}$ is always connected for every polynomial $p$ with degree less than $3$. On the other hand, we show that for each integer $k$ greater than $2$, there exists a polynomial $p$ of degree $k$ such that the spectrum of the Toeplitz operator with symbol ${bar{z}+p}$ has at least one isolated point but has at most finitely many isolated points. Then these results are applied to obtain a class of non-hyponormal Toeplitz operators with bounded harmonic symbols on the Bergman space for which Weyl's theorem holds.
{"title":"The spectral picture of Bergman–Toeplitz operators with harmonic polynomial symbols","authors":"Kunyu Guo, Xianfeng Zhao, Dechao Zheng","doi":"10.4310/arkiv.2023.v61.n2.a5","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n2.a5","url":null,"abstract":"In this paper, it is shown that some new phenomenon related to the spectra of Toeplitz operators with bounded harmonic symbols on the Bergman space. On one hand, we prove that the spectrum of the Toeplitz operator with symbol ${bar{z}+p}$ is always connected for every polynomial $p$ with degree less than $3$. On the other hand, we show that for each integer $k$ greater than $2$, there exists a polynomial $p$ of degree $k$ such that the spectrum of the Toeplitz operator with symbol ${bar{z}+p}$ has at least one isolated point but has at most finitely many isolated points. Then these results are applied to obtain a class of non-hyponormal Toeplitz operators with bounded harmonic symbols on the Bergman space for which Weyl's theorem holds.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135612976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/arkiv.2023.v61.n1.a10
Y. Shitov
{"title":"On local and semi-matching colorings of split graphs","authors":"Y. Shitov","doi":"10.4310/arkiv.2023.v61.n1.a10","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n1.a10","url":null,"abstract":"","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/arkiv.2023.v61.n1.a6
Thi Hien Truong, Nam Trung Tran
{"title":"Regularity of symbolic powers of square-free monomial ideals","authors":"Thi Hien Truong, Nam Trung Tran","doi":"10.4310/arkiv.2023.v61.n1.a6","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n1.a6","url":null,"abstract":"","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70394593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/arkiv.2023.v61.n1.a5
O. Constantin
{"title":"A complex-analytic approach to streamline properties of deep-water Stokes waves","authors":"O. Constantin","doi":"10.4310/arkiv.2023.v61.n1.a5","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n1.a5","url":null,"abstract":"","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70394540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/arkiv.2023.v61.n2.a2
. This paper concerns the overcompleteness of coherent frames for amenable unimodular groups. It is shown that for coherent frames associated with an integrable vector a set of positive Beurling density can be removed yet still leave a frame. The obtained results extend various theorems of [J. Fourier Anal. Appl., 12(3):307-344, 2006] to frames with non-Abelian index sets.
{"title":"Overcompleteness of coherent frames for unimodular amenable groups","authors":"","doi":"10.4310/arkiv.2023.v61.n2.a2","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n2.a2","url":null,"abstract":". This paper concerns the overcompleteness of coherent frames for amenable unimodular groups. It is shown that for coherent frames associated with an integrable vector a set of positive Beurling density can be removed yet still leave a frame. The obtained results extend various theorems of [J. Fourier Anal. Appl., 12(3):307-344, 2006] to frames with non-Abelian index sets.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135659927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/arkiv.2023.v61.n2.a7
Naoki Matsui
{"title":"Minimal-mass blow-up solutions for inhomogeneous nonlinear Schrödinger equations with growing potentials","authors":"Naoki Matsui","doi":"10.4310/arkiv.2023.v61.n2.a7","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n2.a7","url":null,"abstract":"","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135659367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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