<p>Let <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda subset-of double-struck upper R Superscript n">
<mml:semantics>
<mml:mrow>
<mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi>
<mml:mo>⊂<!-- ⊂ --></mml:mo>
<mml:msup>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="double-struck">R</mml:mi>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:msup>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\Lambda \subset \mathbb R^n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> be an algebraic lattice coming from a projective module over the ring of integers of a number field <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K">
<mml:semantics>
<mml:mi>K</mml:mi>
<mml:annotation encoding="application/x-tex">K</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. Let <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Z subset-of double-struck upper R Superscript n">
<mml:semantics>
<mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi class="MJX-tex-caligraphic" mathvariant="script">Z</mml:mi>
</mml:mrow>
<mml:mo>⊂<!-- ⊂ --></mml:mo>
<mml:msup>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="double-struck">R</mml:mi>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:msup>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathcal Z \subset \mathbb R^n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> be the zero locus of a finite collection of polynomials such that <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda neither-a-subset-of-nor-equal-to script upper Z">
<mml:semantics>
<mml:mrow>
<mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi>
<mml:mo>⊈<!-- ⊈ --></mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi class="MJX-tex-caligraphic" mathvariant="script">Z</mml:mi>
</mml:mrow>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\Lambda \nsubseteq \mathcal Z</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> or a finite union of proper full-rank sublattices of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda">
<mml:semantics>
<mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi>
<mml:annotation encoding="application/x-tex">\Lambda</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. Let <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K 1">
<mml:semantics>
<mml:msub>
<mml:mi>K</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:annotation encoding="application/x-tex">K_1</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> be the number field generated over <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K">
<mml:semantics>
<mml:mi>K</mml:mi>
<mml:annotation encoding="application/x-tex">K</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> by coordinates of vectors in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda">
<mml:semantics>
<mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi>
<mml:annotation encoding="application/x-tex">\Lambda</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, and let <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 1 comma ellipsis comma upper L Subscript t Baseline">
<mml:semantics>
<mml:mrow>
<mml:msub>
<mml:mi>L</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>…<!-- … --></mml:mo>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>L</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">L_1,\dots ,L_t</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> be linear forms in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n">
<mml:semantics>
<mml:mi>n</mml:mi>
<mml:annotation encoding="application/x-tex">n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> variables with algebraic coefficients satisfying an appropriate linear independence condition over <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K 1">
<mml:semantics>
<mml:msub>
<mml:mi>K</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:annotation encoding="application/x-tex">K_1</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. For each <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon greater-than 0">
<mml:semantics>
<mml:mrow>
<mml:mi>ε<!-- ε --></mml:mi>
<mml:mo>></mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\varepsilon > 0</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> and <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic a element-of double-struck upper R Superscript n">
<mml:semantics>
<mml:mrow>
<mml:mi mathvariant="bold-italic">a</mml:mi>
<mml:mo>∈<!-- ∈ --></mml:mo>
<mml:msup>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="double-struck">R</mml:mi>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:msup>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\boldsymbol a \in \mathbb R^n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, we prove the existence of a vector <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic x element-of normal upper Lamda minus script upper Z">
<mml:semantics>
<mml:mrow>
<mml:mi mathvariant="bold-italic">x</mml:mi>
<mml:mo>∈<!-- ∈ --></mml:mo>
<mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi>
<mml:mo class="MJX-variant">∖<!-- ∖ --></mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi class="MJX-tex-caligraphic" mathvariant="script">Z</mml:mi>
</mml:mrow>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\boldsymbol x \in \Lambda \setminus \mathcal Z</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> of explicitly bounded sup-norm such that <disp-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper L Subscript i Baseline left-parenthesis bold-italic x right-parenthesis minus a Subscript i Baseline double-vertical-bar greater-than epsilon">
<mml:semantics>
<mml:mrow>
<mml:mo fence="false" stretchy="false">‖<!-- ‖ --></mml:mo>
<mml:msub>
<mml:mi>L</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi mathvariant="bold-italic">x</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>−<!-- − --></mml:mo>
<mml:msub>
<mml:mi>a</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo fence="false" stretchy="false">‖<!-- ‖ --></mml:mo>
<mml:mo>></mml:mo>
<mml:mi>ε<!-- ε --></mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\begin{equation*} \| L_i(\boldsymbol x) - a_i \| > \varepsilon \end{equation*}</mml:annotation>
</mml:semantics>
</mml:math>
</disp-formula>
for each <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-equal-to i less-than-or-equal-to t">
<mml:semantics>
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>≤<!-- ≤ --></mml:mo>
<mml:mi>i</mml:mi>
<mml:mo>≤<!-- ≤ --></mml:mo>
<mml:mi>t</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">1 \leq i \leq t</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, where <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar double-vertical-bar">
<mml:semantics>
<mml:mrow>
<mml:mo fence="false" stretchy="false">‖<!-- ‖ --></mml:mo>
<mml:mtext> </mml:mtext>
<mml:mo fence="false" stretchy="false">‖<!-- ‖ --></mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\|\ \|</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> stands for the distance to the nearest integer. The bound on sup-norm of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic x">
<mml:semantics>
<mml:mi mathvariant="bold-italic">x</mml:mi>
<mml:annotation encoding="application/x-tex">\boldsymbol x</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> depends on <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon">
<mml:semantics>
<mml:mi>ε<!-- ε --></mml:mi>
<mml:annotation encoding="application/x-tex">\varepsilon</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, as well as on <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda">
<mml:semantics>
<mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi>
<mml:annotation e
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