Abstract
This paper proposes the concept of envelope numbers for irrational numbers, divides envelope numbers into upper envelope numbers and lower envelope numbers, and proves several important properties of such numbers. Firstly, the paper gives the uniform distribution theorem of irrational integer multiples. By proving several lemmas of upper envelope numbers and lower envelope numbers, it is proved that there are countless upper envelope numbers and lower envelope numbers of irrational numbers. At the same time, it is proved that the sum of the maximum upper envelope number and lower envelope number that does not exceed a given positive integer is also an envelope number.
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