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<title>Generalized arithmetic progression</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Generalized arithmetic progression</span></span>

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<p><br>

In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>generalized arithmetic progression</b> (or <b>multiple arithmetic progression</b>) is a generalization of an <a href="Arithmetic_progression" title="Arithmetic progression">arithmetic progression</a> equipped with multiple common differences – whereas an arithmetic progression is generated by a single common difference, a generalized arithmetic progression can be generated by multiple common differences. For example, the <a href="Sequence" title="Sequence">sequence</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 17,20,22,23,25,26,27,28,29,\dots }">

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<annotation encoding="application/x-tex">{\displaystyle 17,20,22,23,25,26,27,28,29,\dots }</annotation>

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</math></span><img src="./c7d4c7c6021263327e3306cde087f75716723615.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.953ex; height:2.509ex;" alt="{\displaystyle 17,20,22,23,25,26,27,28,29,\dots }" loading="lazy"></span> is not an arithmetic progression, but is instead generated by starting with 17 and adding either 3 <i>or</i> 5, thus allowing multiple common differences to generate it.

A <b>semilinear set</b> generalizes this idea to multiple dimensions – it is a set of vectors of integers, rather than a set of integers.

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<div class="mw-heading mw-heading2"><h2 id="Finite_generalized_arithmetic_progression">Finite generalized arithmetic progression</h2></div>

<p>A <b>finite generalized arithmetic progression</b>, or sometimes just <b>generalized arithmetic progression (GAP)</b>, of dimension <i>d</i> is defined to be a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of the form

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<annotation encoding="application/x-tex">{\displaystyle \{x_{0}+\ell _{1}x_{1}+\cdots +\ell _{d}x_{d}:0\leq \ell _{1}<L_{1},\ldots ,0\leq \ell _{d}<L_{d}\}}</annotation>

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</math></span><img src="./08b0d2b89b5aa67f4af8ea15d160dafd99104215.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.076ex; height:2.843ex;" alt="{\displaystyle \{x_{0}+\ell _{1}x_{1}+\cdots +\ell _{d}x_{d}:0\leq \ell _{1}<L_{1},\ldots ,0\leq \ell _{d}<L_{d}\}}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0},x_{1},\dots ,x_{d},L_{1},\dots ,L_{d}\in \mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle x_{0},x_{1},\dots ,x_{d},L_{1},\dots ,L_{d}\in \mathbb {Z} }</annotation>

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</math></span><img src="./86e5ba6906105980faae5f8a49ca8e332c1ebba0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.317ex; height:2.509ex;" alt="{\displaystyle x_{0},x_{1},\dots ,x_{d},L_{1},\dots ,L_{d}\in \mathbb {Z} }" loading="lazy"></span>. The product <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{1}L_{2}\cdots L_{d}}">

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<annotation encoding="application/x-tex">{\displaystyle L_{1}L_{2}\cdots L_{d}}</annotation>

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</math></span><img src="./49098e4499ae885212f0542f4d1323eb0c8eca3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.446ex; height:2.509ex;" alt="{\displaystyle L_{1}L_{2}\cdots L_{d}}" loading="lazy"></span> is called the <b>size</b> of the generalized arithmetic progression; the <a href="Cardinality" title="Cardinality">cardinality</a> of the set can differ from the size if some elements of the set have multiple representations. If the cardinality equals the size, the progression is called <b>proper</b>. Generalized arithmetic progressions can be thought of as a projection of a higher dimensional grid into <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

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</math></span><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"></span>. This projection is <a href="Injective" class="mw-redirect" title="Injective">injective</a> if and only if the generalized arithmetic progression is proper.

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<div class="mw-heading mw-heading2"><h2 id="Semilinear_sets">Semilinear sets</h2></div>

<p>Formally, an arithmetic progression of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ^{d}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ^{d}}</annotation>

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</math></span><img src="./699f5951d283dce31055fdec7c32b34cdebf297f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {N} ^{d}}" loading="lazy"></span> is an infinite sequence of the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} ,\mathbf {v} +\mathbf {v} ',\mathbf {v} +2\mathbf {v} ',\mathbf {v} +3\mathbf {v} ',\ldots }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ,\mathbf {v} +\mathbf {v} ',\mathbf {v} +2\mathbf {v} ',\mathbf {v} +3\mathbf {v} ',\ldots }</annotation>

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</math></span><img src="./23d80a6d763f406ee3fc2c90c03d4836321f777f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.636ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} ,\mathbf {v} +\mathbf {v} ',\mathbf {v} +2\mathbf {v} ',\mathbf {v} +3\mathbf {v} ',\ldots }" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation>

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</math></span><img src="./35c1866e359fbfd2e0f606c725ba5cc37a5195d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} '}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} '}</annotation>

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</math></span><img src="./eda93d5afe45eb64a74e252ad598abad1531e244.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.096ex; height:2.509ex;" alt="{\displaystyle \mathbf {v} '}" loading="lazy"></span> are fixed vectors in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ^{d}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ^{d}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ^{d}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \left\{\mathbf {v} +\sum _{i=1}^{m}k_{i}\mathbf {v} _{i}\,\colon \,k_{1},\dots ,k_{m}\in \mathbb {N} \right\},}</annotation>

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</math></span><img src="./214d03e0407de9ae2208e2976914b420c204c662.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.652ex; height:7.509ex;" alt="{\displaystyle \left\{\mathbf {v} +\sum _{i=1}^{m}k_{i}\mathbf {v} _{i}\,\colon \,k_{1},\dots ,k_{m}\in \mathbb {N} \right\},}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ,\mathbf {v} _{1},\dots ,\mathbf {v} _{m}}</annotation>

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</math></span><img src="./8ba164f6d850d8e212a3d5fbdddbbd36deca76aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.174ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} ,\mathbf {v} _{1},\dots ,\mathbf {v} _{m}}" loading="lazy"></span> are fixed vectors in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ^{d}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ^{d}}</annotation>

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</math></span><img src="./699f5951d283dce31055fdec7c32b34cdebf297f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {N} ^{d}}" loading="lazy"></span>. A subset of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ^{d}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ^{d}}</annotation>

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</math></span><img src="./699f5951d283dce31055fdec7c32b34cdebf297f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {N} ^{d}}" loading="lazy"></span> is said to be <b>semilinear</b> if it is a finite <a href="Union_(set_theory)" title="Union (set theory)">union</a> of linear sets.

</p><p>The semilinear sets are exactly the sets definable in <a href="Presburger_arithmetic" title="Presburger arithmetic">Presburger arithmetic</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<ul><li><a href="Freiman's_theorem" title="Freiman's theorem">Freiman's theorem</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFGinsburgSpanier1966" class="citation journal cs1">Ginsburg, Seymour; Spanier, Edwin Henry (1966). <a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Fpjm.1966.16.285">"Semigroups, Presburger Formulas, and Languages"</a>. <i>Pacific Journal of Mathematics</i>. <b>16</b> (2): <span class="nowrap">285–</span>296. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Fpjm.1966.16.285">10.2140/pjm.1966.16.285</a></span>.</cite></span>

</li>

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<ul><li><cite id="CITEREFNathanson1996" class="citation book cs1">Nathanson, Melvyn B. (1996). <i>Additive Number Theory: Inverse Problems and Geometry of Sumsets</i>. <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>. Vol. 165. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-94655-1</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0859.11003">0859.11003</a>.</cite></li></ul></div><div><div class="zim-footer">

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