A function f ： N → Ris called additive if f（mn） = f（m） ＋f （n） for all m, n with （m, n） = 1. Let μ（x） = maxn≤x（f（n） - f（n ＋ 1）） and v（x） = maxn≤x（f（n ＋ 1） - f（n））. In 1979, Ruzsa proved ...A function f ： N → Ris called additive if f（mn） = f（m） ＋f （n） for all m, n with （m, n） = 1. Let μ（x） = maxn≤x（f（n） - f（n ＋ 1）） and v（x） = maxn≤x（f（n ＋ 1） - f（n））. In 1979, Ruzsa proved that there exists a constant c such that for any additive function f, μ（x） ≤ cv（x^2） ＋ cf, where cf is a constant depending only on f. Denote by Raf the least such constant c. We cMl Raf Ruzsa＇s constant on additive functions. In this paper, we prove that Raf≤ 20.展开更多
For A Zm and n ∈ Zm, let σA（n） be the number of solutions of equation n =x ＋ y, x, y ∈ A. Given a positive integer m, let Rm be the least positive integer r such that there exists a set A Zm with A ＋ A = Zm ...For A Zm and n ∈ Zm, let σA（n） be the number of solutions of equation n =x ＋ y, x, y ∈ A. Given a positive integer m, let Rm be the least positive integer r such that there exists a set A Zm with A ＋ A = Zm and σA（n） ≤ r. Recently, Chen Yonggao proved that all Rm ≤ 288. In this paper, we obtain new upper bounds of some special type Rkp2.展开更多
Given an infinite group G, we consider the finitely additive invariant measure defined on finite unions of cosets of finite index subgroups. We show that this shares many properties with the size of subsets of a finit...Given an infinite group G, we consider the finitely additive invariant measure defined on finite unions of cosets of finite index subgroups. We show that this shares many properties with the size of subsets of a finite group, for instance we can obtain equivalent results on the Ruzsa distance and product free sets. In particular, if G has infinitely many finite index subgroups, then it has subsets S of measure arbitrarily close to 1/2 with square S2 having measure less than 1. We also examine properties of the Ruzsa distance on the set of finite index subgroups of an infinite group, whereupon it becomes a genuine metric.展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.11071121and11126302)supported by Natural Science Foundation of the Jiangsu Higher Education Institutions(GrantNo.11KJB110006) the Foundation of Nanjing University of Information Science&Technology(Grant No.20110421)
文摘A function f ： N → Ris called additive if f（mn） = f（m） ＋f （n） for all m, n with （m, n） = 1. Let μ（x） = maxn≤x（f（n） - f（n ＋ 1）） and v（x） = maxn≤x（f（n ＋ 1） - f（n））. In 1979, Ruzsa proved that there exists a constant c such that for any additive function f, μ（x） ≤ cv（x^2） ＋ cf, where cf is a constant depending only on f. Denote by Raf the least such constant c. We cMl Raf Ruzsa＇s constant on additive functions. In this paper, we prove that Raf≤ 20.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 10901002 10771103)
文摘For A Zm and n ∈ Zm, let σA（n） be the number of solutions of equation n =x ＋ y, x, y ∈ A. Given a positive integer m, let Rm be the least positive integer r such that there exists a set A Zm with A ＋ A = Zm and σA（n） ≤ r. Recently, Chen Yonggao proved that all Rm ≤ 288. In this paper, we obtain new upper bounds of some special type Rkp2.
文摘Given an infinite group G, we consider the finitely additive invariant measure defined on finite unions of cosets of finite index subgroups. We show that this shares many properties with the size of subsets of a finite group, for instance we can obtain equivalent results on the Ruzsa distance and product free sets. In particular, if G has infinitely many finite index subgroups, then it has subsets S of measure arbitrarily close to 1/2 with square S2 having measure less than 1. We also examine properties of the Ruzsa distance on the set of finite index subgroups of an infinite group, whereupon it becomes a genuine metric.
