On smooth sets of integers

This work studies evenly distributed sets of integers—sets whose quantity within each interval is proportional to the size of the interval, up to a bounded additive deviation. Namely, for ρ , Δ ∈ R a set A of integers is ( ρ , Δ ) - smooth if abs ( | I | ⋅ ρ − | I ∩ A | ) < Δ for any interval I of integers; a set A is Δ -smooth if it is ( ρ , Δ ) -smooth for some real number ρ . The paper introduces the concept of Δ -smooth sets and studies their mathematical structure. It focuses on tools for constructing smooth sets having certain desirable properties and, in particular, on mathematical operations on these sets. Three additional papers by us are build on the work of this paper and present practical applications of smooth sets to common and well-studied scheduling problems. the above mathematical operations is composition of sets of natural numbers. For two infinite sets A , B ⊆ N , the composition of A and B is the subset D of A such that, for all i , the i th member of A is in D if and only if the i th member of N is in B . This operator enables the partition of a ( ρ , Δ ) -smooth set into two sets that are ( ρ 1 , Δ ) -smooth and ( ρ 2 , Δ ) -smooth, for any ρ 1 , ρ 2 and Δ obeying some reasonable restrictions. Another powerful tool for constructing smooth sets is a one-to-one partial function f from the unit interval into the natural numbers having the property that any real interval X ⊆ [ 0 , 1 ) has a subinterval Y which is ‘very close’ to X s.t.  f ( Y ) is ( ρ , Δ ) -smooth, where ρ is the length of Y and Δ is a small constant.