Submultiplication and Growth Rate of Growth Functions on the Integer Group

本文给出了整数群 Z 上一个真的长度函数，并证明了这个长度函数诱导的增长函数不是次可乘的， 但却满足一个一般的次可乘不等式。同时计算出了该增长函数的增长率是欧拉数 e。We give a proper length function on the integer group Z, and demonstrate that its induced growth function is not submultiplicative, but satisfies a general submultiplica- tive inequality. We also show that the growth rate of this growth function is Euler’s number e.