Union of Sets of Lengths of Numerical Semigroups

Let <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><mo>=</mo><mo>〈</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>a</mi><mi>p</mi></msub><mo>〉</mo></mrow></semantics></math></inline-formula> be a numerical semigroup, let <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></semantics></math></inline-formula> and let <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="sans-serif">Z</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> be its set of factorizations. The set of lengths is denoted by <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">L</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><mo>{</mo><mi mathvariant="monospace">L</mi><mrow><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>x</mi><mi>p</mi></msub><mo>)</mo></mrow><mo>∣</mo><mrow><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>x</mi><mi>p</mi></msub><mo>)</mo></mrow><mo>∈</mo><mi mathvariant="sans-serif">Z</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>}</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="monospace">L</mi><mrow><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>x</mi><mi>p</mi></msub><mo>)</mo></mrow><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>x</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula>. The following sets can then be defined: <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="sans-serif">W</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>s</mi><mo>∈</mo><mi>S</mi><mo>∣</mo><mo>∃</mo><mi>x</mi><mo>∈</mo><mi mathvariant="sans-serif">Z</mi><mo>(</mo><mi>s</mi><mo>)</mo><mrow><mi>such</mi><mi>that</mi></mrow><mi mathvariant="monospace">L</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>n</mi><mo>}</mo></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>ν</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>⋃</mo><mrow><mi>s</mi><mo>∈</mo><mi mathvariant="sans-serif">W</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msub><mi mathvariant="script">L</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><msub><mi>l</mi><mn>1</mn></msub><mo><</mo><msub><mi>l</mi><mn>2</mn></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mi>l</mi><mi>r</mi></msub><mo>}</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="sans-serif">Δ</mi><mi>ν</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><msub><mi>l</mi><mn>2</mn></msub><mo>−</mo><msub><mi>l</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>l</mi><mi>r</mi></msub><mo>−</mo><msub><mi>l</mi><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. In this paper, we prove that the function <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="sans-serif">Δ</mi><mi>ν</mi><mo>:</mo><mi mathvariant="double-struck">N</mi><mo stretchy="false">→</mo><mi mathvariant="script">P</mi><mo>(</mo><mi mathvariant="double-struck">N</mi><mo>)</mo></mrow></semantics></math></inline-formula> is almost periodic with period <inline-formula><math display="inline"><semantics><mrow><mi>lcm</mi><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mi>p</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>.