دراسة اعداد بيزو في المجموعة S^3 n S0 و اصغر عنصر في المجموعة S^3

In this work we have proved Grandet-Pisot conjecture which says that the smallest element of the set is ,and some other results related to the elements of the sets and. We have proved that and that and we deduced that the smallest element of the set is . Furthermore , we have determined some special sets of fractional functions which belong to and and definite elements from the two sets and .