【摘要】 如果实数对<α,β>对于任意的使‖n_kα‖→0的整数序列 n₁,n₂,…皆有‖n_kβ‖→0,则称<α,β>为一个 Kátai 对.这里‖‖表示实数与最近的整数的距离.I.Káitai 在[1],[2]中曾指出:若<α,β>为一个 Kátai 对,则β为整数或为α的整数倍.本文认为这一结论欠妥.并用一种新颖的初等方法证明了:实数对<α,β>是 Kátai 对的充分必要条件为存在整数 m,n 使β=mα+n.还就 I.Kátai 关于积性函数的一个重要结论给出了修正证明.更多还原

【Abstract】 A real number pair<α,β>is called a Kátai-pair if the relation‖n_kβ‖→0 holds for every sequence n₁,n₂,…of integers satisfying‖n_kα‖→0 ,where ‖‖denotes the distance from a real number to the nearest integer.In[1]and[2]I.Kátai had ever used an assertion:If<α,β>is a Kátai-pair,then β is an integer,or β=kα,with a stutable integer k.This paper points out that this assertion is incorrect,and proves the following result:a real number pair<α,β>is a Kátai-pair iff there exist integers m,n such that β =ma+n.Using this result author give a correct proof of an important assertion which was given by I.Kátai in[1]about the multiplicative functions.更多还原