<p>Let <i>C</i> and <i>W</i> be two integer sets. If <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C+W={\mathbb {Z}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo>+</mo> <mi>W</mi> <mo>=</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation>, then we say that <i>C</i> is an additive complement to <i>W</i>. If no proper subset of <i>C</i> is an additive complement to <i>W</i>, then we say that <i>C</i> is a minimal additive complement to <i>W</i>. Let <i>c</i> and <i>d</i> be two positive integers with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(c\mid d\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>∣</mo> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we prove that there exists an infinite, not eventually periodic set <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(W\subset {\mathbb {N}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mo>⊂</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(w_{i+1}-w_{i}\in \{c,d\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>w</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>w</mi> <mi>i</mi> </msub> <mo>∈</mo> <mrow> <mo stretchy="false">{</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <i>i</i> and there exists a minimal complement to <i>W</i>. Moreover, we partially solve a problem of Kiss, Sándor and Yang.</p>