Let G be a finite abelian group and p be the smallest prime dividing |G|. Let S be a sequence over G. We say that S is regular if for every proper subgroup H ( G, S contains at most |H | − 1 terms from H . Let c0(G) be the smallest integer t such that every regular sequence S over G of length |S| ≥ t forms an additive basis of G, i.e., ∑ (S) = G. The invariant c0(G) was first studied by Olson and Peng in 1980’s, and since then it has been determined for all finite abelian groups except for the groups with rank 2 and a few groups of rank 3 or 4 with order less than 10. In this paper, we focus on the remaining case concerning groups of rank 2. It was conjectured by the first author and Han (Int. J. Number Theory 13 (2017) 2453-2459) that c0(G) = pn+2p− 3 where G = Cp ⊕ Cpn with n ≥ 3. We confirm the conjecture for the case when p = 3 and n = q (≥ 5) is a prime number.
设G是有限阿贝尔群，p是最小素数除|G|。设S是G上的一个序列。我们说S是正则的，如果对于每个适当子群H（G，S最多包含来自H的|H|−1项。设c0（G）是最小整数t，使得每个长度|S|≥t的G上的正则序列S形成G的加性基，即∑（S）=G。不变量c0（G）最早由Olson和Peng在20世纪80年代研究，并且从那时起，已经确定了除了秩为2的组和秩为3或4且阶小于10的少数组之外的所有有限阿贝尔组。在本文中，我们关注关于秩为2的群的剩余情况。第一作者和Han（Int.J.Number Theory 13（2017）2453-2459）推测c0（G）=pn+2p−3，其中G=CpŞCpn，n≥3。我们证实了当p=3和n=q（≥5）是素数时的猜想。
{"title":"Additive bases of $C_3oplus C_{3q}$","authors":"Yongke Qu, Yuanlin Li","doi":"10.4064/cm8515-6-2021","DOIUrl":"https://doi.org/10.4064/cm8515-6-2021","url":null,"abstract":"Let G be a finite abelian group and p be the smallest prime dividing |G|. Let S be a sequence over G. We say that S is regular if for every proper subgroup H ( G, S contains at most |H | − 1 terms from H . Let c0(G) be the smallest integer t such that every regular sequence S over G of length |S| ≥ t forms an additive basis of G, i.e., ∑ (S) = G. The invariant c0(G) was first studied by Olson and Peng in 1980’s, and since then it has been determined for all finite abelian groups except for the groups with rank 2 and a few groups of rank 3 or 4 with order less than 10. In this paper, we focus on the remaining case concerning groups of rank 2. It was conjectured by the first author and Han (Int. J. Number Theory 13 (2017) 2453-2459) that c0(G) = pn+2p− 3 where G = Cp ⊕ Cpn with n ≥ 3. We confirm the conjecture for the case when p = 3 and n = q (≥ 5) is a prime number.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49301244","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}
Let G be a multiplicatively written finite group. We denote by E(G) the smallest integer t such that every sequence of t elements in G contains a product-one subsequence of length |G|. In 1961, Erdős, Ginzburg and Ziv proved that E(G) ≤ 2|G|−1 for every finite ablian group G and this result is known as the Erdős-Ginzburg-Ziv Theorem. In 2005, Zhuang and Gao conjectured that E(G) = d(G) + |G|, where d(G) is the small Davenport constant. In this paper, we confirm the conjecture for the case when G = 〈x, y|x = y = 1, xyx = y〉, where p is the smallest prime divisor of |G| and gcd(p(r − 1),m) = 1.
{"title":"On a conjecture of Zhuang and Gao","authors":"Yongke Qu, Yuanlin Li","doi":"10.4064/cm8685-2-2022","DOIUrl":"https://doi.org/10.4064/cm8685-2-2022","url":null,"abstract":"Let G be a multiplicatively written finite group. We denote by E(G) the smallest integer t such that every sequence of t elements in G contains a product-one subsequence of length |G|. In 1961, Erdős, Ginzburg and Ziv proved that E(G) ≤ 2|G|−1 for every finite ablian group G and this result is known as the Erdős-Ginzburg-Ziv Theorem. In 2005, Zhuang and Gao conjectured that E(G) = d(G) + |G|, where d(G) is the small Davenport constant. In this paper, we confirm the conjecture for the case when G = 〈x, y|x = y = 1, xyx = y〉, where p is the smallest prime divisor of |G| and gcd(p(r − 1),m) = 1.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48460074","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}
This paper gives new two-weight bump conditions for the sparse operators related to iterated commutators of fractional integrals. As applications, the two-weight bounds for iterated commutators of fractional integrals under more general bump conditions are obtained. Meanwhile, the necessity of two-weight bump conditions as well as the converse of Bloom type estimates for iterated commutators of fractional integrals are also given.
{"title":"Bump conditions and two-weight inequalities for commutators of fractional integrals","authors":"Yongming Wen, Huo-xiong Wu","doi":"10.4064/cm8703-4-2022","DOIUrl":"https://doi.org/10.4064/cm8703-4-2022","url":null,"abstract":"This paper gives new two-weight bump conditions for the sparse operators related to iterated commutators of fractional integrals. As applications, the two-weight bounds for iterated commutators of fractional integrals under more general bump conditions are obtained. Meanwhile, the necessity of two-weight bump conditions as well as the converse of Bloom type estimates for iterated commutators of fractional integrals are also given.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44459512","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}
． 让s (n) = (cid): 80) d | n, d < n d合适divisors》denote the sum of n。自然是每到conjecture that for《equivalence n是整数k≥2,k th powerfree⇐⇒s (n)是k th powerfree珍藏几乎总是(asymptotic密度之意义,on a组1)。我们证明这个for k≥4。
{"title":"Powerfree sums of proper divisors","authors":"P. Pollack, A. Roy","doi":"10.4064/cm8616-10-2021","DOIUrl":"https://doi.org/10.4064/cm8616-10-2021","url":null,"abstract":". Let s ( n ) := (cid:80) d | n,d<n d denote the sum of the proper divisors of n . It is natural to conjecture that for each integer k ≥ 2 , the equivalence n is k th powerfree ⇐⇒ s ( n ) is k th powerfree holds almost always (meaning, on a set of asymptotic density 1 ). We prove this for k ≥ 4 .","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46099791","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}
It is known that G-functions solutions of a linear differential equation of order 1 with coefficients in Q(z) are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we determine the form of a G-function solution of an inhomogeneous equation of order 1 with coefficients in Q(z), as well as that of a G-function f of differential order 2 over Q(z) and such that f and f ′ are algebraically dependent over C(z). Our results apply more generally to holonomic Nilsson-Gevrey arithmetic series of order 0 that encompass G-functions.
{"title":"A note on $G$-operators of order $2$","authors":"S. Fischler, T. Rivoal","doi":"10.4064/cm8600-3-2022","DOIUrl":"https://doi.org/10.4064/cm8600-3-2022","url":null,"abstract":"It is known that G-functions solutions of a linear differential equation of order 1 with coefficients in Q(z) are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we determine the form of a G-function solution of an inhomogeneous equation of order 1 with coefficients in Q(z), as well as that of a G-function f of differential order 2 over Q(z) and such that f and f ′ are algebraically dependent over C(z). Our results apply more generally to holonomic Nilsson-Gevrey arithmetic series of order 0 that encompass G-functions.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46325998","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}
. In this paper, we study the k -th moment of central values of the family of quadratic Dirichlet L -functions of moduli 8 p , with p ranging over odd primes. Assuming the truth of the generalized Riemann hypothesis, we establish sharp upper and lower bounds for the k -th power moment of these L -values for all real k ≥ 0.
． 本文研究了模为8p的二次Dirichlet L -函数族中心值的k阶矩，其中p的取值范围为奇数素数。在广义黎曼假设成立的前提下，我们为所有实k≥0时这些L值的k次幂矩建立了清晰的上界和下界。
{"title":"Bounds for moments of quadratic Dirichlet $L$-functions of prime-related moduli","authors":"Peng Gao, Liangyi Zhao","doi":"10.4064/cm8650-1-2022","DOIUrl":"https://doi.org/10.4064/cm8650-1-2022","url":null,"abstract":". In this paper, we study the k -th moment of central values of the family of quadratic Dirichlet L -functions of moduli 8 p , with p ranging over odd primes. Assuming the truth of the generalized Riemann hypothesis, we establish sharp upper and lower bounds for the k -th power moment of these L -values for all real k ≥ 0.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44228415","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}
In spaces of metrics, we investigate topological distributions of the doubling property, the uniform disconnectedness, and the uniform perfectness, which are the quasi-symmetrically invariant properties appearing in the David--Semmes theorem. We show that the set of all doubling metrics and the set of all uniformly disconnected metrics are dense in spaces of metrics on finite-dimensional and zero-dimensional compact metrizable spaces, respectively. Conversely, this denseness of the sets implies the finite-dimensionality, zero-dimensionality, and the compactness of metrizable spaces. We also determine the topological distribution of the set of all uniformly perfect metrics in the space of metrics on the Cantor set.
{"title":"On dense subsets in spaces of metrics","authors":"Yoshito Ishiki","doi":"10.4064/cm8580-9-2021","DOIUrl":"https://doi.org/10.4064/cm8580-9-2021","url":null,"abstract":"In spaces of metrics, we investigate topological distributions of the doubling property, the uniform disconnectedness, and the uniform perfectness, which are the quasi-symmetrically invariant properties appearing in the David--Semmes theorem. We show that the set of all doubling metrics and the set of all uniformly disconnected metrics are dense in spaces of metrics on finite-dimensional and zero-dimensional compact metrizable spaces, respectively. Conversely, this denseness of the sets implies the finite-dimensionality, zero-dimensionality, and the compactness of metrizable spaces. We also determine the topological distribution of the set of all uniformly perfect metrics in the space of metrics on the Cantor set.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43096608","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}
We investigate the geometry of $C(K,X)$ and $ell_{infty}(X)$ spaces through complemented subspaces of the form $left(bigoplus_{iin varGamma}X_iright)_{c_0}$. Concerning the geometry of $C(K,X)$ spaces we extend some results of D. Alspach and E. M. Galego from cite{AlspachGalego}. On $ell_{infty}$-sums of Banach spaces we prove that if $ell_{infty}(X)$ has a complemented subspace isomorphic to $c_0(Y)$, then, for some $n in mathbb{N}$, $X^n$ has a subspace isomorphic to $c_0(Y)$. We further prove the following: (1) If $C(K)sim c_0(C(K))$ and $C(L)sim c_0(C(L))$ and $ell_{infty}(C(K))sim ell_{infty}(C(L))$, then $K$ and $L$ have the same cardinality. (2) If $K_1$ and $K_2$ are infinite metric compacta, then $ell_{infty}(C(K_1))sim ell_{infty}(C(K_2))$ if and only if $C(K_1)$ is isomorphic to $C(K_2)$.
我们通过$left(bigoplus_{iin varGamma}X_iright)_{c_0}$形式的互补子空间研究了$C(K,X)$和$ell_{infty}(X)$空间的几何。关于$C(K,X)$空间的几何性质，我们推广了D. Alspach和E. M. Galego在cite{AlspachGalego}上的一些结果。在Banach空间的$ell_{infty}$ -和上，证明了如果$ell_{infty}(X)$具有与$c_0(Y)$同构的补子空间，则对于某些$n in mathbb{N}$, $X^n$具有与$c_0(Y)$同构的子空间。我们进一步证明了:(1)如果$C(K)sim c_0(C(K))$与$C(L)sim c_0(C(L))$和$ell_{infty}(C(K))sim ell_{infty}(C(L))$，则$K$和$L$具有相同的基数。(2)如果$K_1$和$K_2$是无限度量紧致，则$ell_{infty}(C(K_1))sim ell_{infty}(C(K_2))$当且仅当$C(K_1)$同构于$C(K_2)$。
{"title":"Complementations in $C(K,X)$ and $ell _infty (X)$","authors":"Leandro Candido","doi":"10.4064/cm8868-10-2022","DOIUrl":"https://doi.org/10.4064/cm8868-10-2022","url":null,"abstract":"We investigate the geometry of $C(K,X)$ and $ell_{infty}(X)$ spaces through complemented subspaces of the form $left(bigoplus_{iin varGamma}X_iright)_{c_0}$. Concerning the geometry of $C(K,X)$ spaces we extend some results of D. Alspach and E. M. Galego from cite{AlspachGalego}. On $ell_{infty}$-sums of Banach spaces we prove that if $ell_{infty}(X)$ has a complemented subspace isomorphic to $c_0(Y)$, then, for some $n in mathbb{N}$, $X^n$ has a subspace isomorphic to $c_0(Y)$. We further prove the following: \u0000(1) If $C(K)sim c_0(C(K))$ and $C(L)sim c_0(C(L))$ and $ell_{infty}(C(K))sim ell_{infty}(C(L))$, then $K$ and $L$ have the same cardinality. \u0000(2) If $K_1$ and $K_2$ are infinite metric compacta, then $ell_{infty}(C(K_1))sim ell_{infty}(C(K_2))$ if and only if $C(K_1)$ is isomorphic to $C(K_2)$.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41882649","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}
We exhibit $88$ examples of rank zero elliptic curves over the rationals with $|sza(E)| > 63408^2$, which was the largest previously known value for any explicit curve. Our record is an elliptic curve $E$ with $|sza(E)| = 1029212^2 = 2^4cdot 79^2 cdot 3257^2$. We can use deep results by Kolyvagin, Kato, Skinner-Urban and Skinner to prove that, in some cases, these orders are the true orders of $sza$. For instance, $410536^2$ is the true order of $sza(E)$ for $E= E_4(21,-233)$ from the table in section 2.3.
{"title":"Elliptic curves with exceptionally large analytic order of the Tate–Shafarevich groups","authors":"A. Dkabrowski, L. Szymaszkiewicz","doi":"10.4064/CM8008-9-2020","DOIUrl":"https://doi.org/10.4064/CM8008-9-2020","url":null,"abstract":"We exhibit $88$ examples of rank zero elliptic curves over the rationals with $|sza(E)| > 63408^2$, which was the largest previously known value for any explicit curve. Our record is an elliptic curve $E$ with $|sza(E)| = 1029212^2 = 2^4cdot 79^2 cdot 3257^2$. We can use deep results by Kolyvagin, Kato, Skinner-Urban and Skinner to prove that, in some cases, these orders are the true orders of $sza$. For instance, $410536^2$ is the true order of $sza(E)$ for $E= E_4(21,-233)$ from the table in section 2.3.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46148455","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}
We prove that any one-relator group $G$ is the fundamental group of a compact Sasakian manifold if and only if $G$ is either finite cyclic or isomorphic to the fundamental group of a compact Riemann surface of genus g > 0 with at most one orbifold point of order $n geq 1$. We also classify all groups of deficiency at least two that are also the fundamental group of some compact Sasakian manifold.
{"title":"One-relator Sasakian groups","authors":"I. Biswas, Mahan Mj","doi":"10.4064/CM8521-3-2021","DOIUrl":"https://doi.org/10.4064/CM8521-3-2021","url":null,"abstract":"We prove that any one-relator group $G$ is the fundamental group of a compact Sasakian manifold if and only if $G$ is either finite cyclic or isomorphic to the fundamental group of a compact Riemann surface of genus g > 0 with at most one orbifold point of order $n geq 1$. We also classify all groups of deficiency at least two that are also the fundamental group of some compact Sasakian manifold.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48612998","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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