Union of Sets of Lengths of Numerical Semigroups

Let S = 〈 a 1 , … , a p 〉 be a numerical semigroup, let s ∈ S and let Z ( s ) be its set of factorizations. The set of lengths is denoted by L ( s ) = { L ( x 1 , ⋯ , x p ) ∣ ( x 1 , ⋯ , x p ) ∈ Z ( s ) } , where L ( x 1 , ⋯ , x p ) = x 1 + ⋯ + x p . The following sets can then be defined: W ( n ) = { s ∈ S ∣ ∃ x ∈ Z ( s ) such that L ( x ) = n } , ν ( n ) = ⋃ s ∈ W ( n ) L ( s ) = { l 1 < l 2 < ⋯ < l r } and Δ ν ( n ) = { l 2 − l 1 , … , l r − l r − 1 } . In this paper, we prove that the function Δ ν : N → P ( N ) is almost periodic with period lcm ( a 1 , a p ) .