Real numbers with no unusually-accurate dyadic rational approximations. The red circles surround the real numbers that are approximated to within error <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{6}}/2^{i}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="false" scriptlevel="0"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mi>i</mi></mrow></msup></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{6}}/2^{i}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d66d2a0b4287274ce6d90bbbf6479bb0174ffe6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.338ex; width:4.783ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{6}}/2^{i}}"></span> by a dyadic rational of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n/2^{i}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mi>i</mi></mrow></msup></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle n/2^{i}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b52793068f7b4b257dace031b719a82aeb891f6a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.519ex; height:3.176ex;" alt="{\displaystyle n/2^{i}}"></span> For numbers in the fractal Cantor set outside the circles, all dyadic rational approximations have larger errors.
<a href="//commons.wikimedia.org/wiki/User:David_Eppstein" class="mw-redirect" title="User:David Eppstein">David Eppstein</a>
<span class="int-own-work" lang="en">Own work</span>