Let A and B be two subsets of the set of all non-negative
integers with 0 ε A and O ε B. The sum of the sets A and B is
the set C = A + B = {a + b: a ε A, b ε B). For n...
This thesis brings together under one cover a survey of the
history of the real number pi along with a study of the present state
of its theory, application, and calculation.
In 1932 A. Ya. Khinchin gave the first partial solution of the celebrated 1931 αβ Conjecture of L.G. Schnirelmann and E. Landau of the density of sums of sets on integers, which was completely proved in 1942 by H.B. Mann.
Khinchin's theorem is proved along with theorems of P. Scherk...
This thesis contains a collection of summation formulas for
the greatest integer part function. Proofs are supplied for original
results and for those formulas which are stated without proof in the
literature. References are given for formulas and proofs which
appear in the literature.
In this thesis, we study conditions not involving density which
guarantee that a given positive integer is contained in a sum of sets
of nonnegative integers. We survey the literature, give more detailed
proofs of some known theorems, develop some new theorems,
and make some conjectures.