Constructing optimal star discrepancy sets

We provide in this paper a constructive proof of optimal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">L_{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> star discrepancy values in dimension 2 for up to 21 points and up to 8 points in dimension 3. This extends work by White (Numer. Math. <bold>27</bold> (1976/77), no. 2, 157–164) for up to six points in dimension 2 and of Larcher and Pillichshammer (J. Comput. Appl. Math. <bold>206</bold> (2007), no. 2, 977–985) for two points in arbitrary dimensions. We show that these optimal sets have a far lower discrepancy than the previous references and, perhaps more importantly, present a very different structure.