Editing Smith–Volterra–Cantor set (section)

== Other fat Cantor sets ==

In general, one can remove <math>r_n</math> from each remaining subinterval at the <math>n</math>th step of the algorithm, and end up with a Cantor-like set.  The resulting set will have positive measure if and only if the sum of the sequence is less than the measure of the initial interval. For instance, suppose the middle intervals of length <math>a^n</math> are removed from <math>[0, 1]</math> for each <math>n</math>th iteration, for some <math>0 \leq a \leq \dfrac{1}{3}.</math> Then, the resulting set has Lebesgue measure 
<math display=block>\begin{align}
1 - \sum _{n=0}^\infty 2^n a ^ {n+1} &= 1 - a \sum _{n=0}^\infty (2a)^n \\[5pt]
                                     &= 1 - a \frac{1}{1 - 2a} \\[5pt]
                                     &= \frac{1 - 3a}{1 - 2a}
\end{align}
</math>
which goes from <math>0</math> to <math>1</math> as <math>a</math> goes from <math>1/3</math> to <math>0.</math>  (<math>a > 1/3</math> is impossible in this construction.)

Cartesian products of Smith–Volterra–Cantor sets can be used to find [[totally disconnected space|totally disconnected set]]s in higher dimensions with nonzero measure. By applying the [[Denjoy–Riesz theorem]] to a two-dimensional set of this type, it is possible to find an [[Osgood curve]], a [[Jordan curve]] such that the points on the curve have positive area.<ref>{{citation
 | last1 = Balcerzak | first1 = M.
 | last2 = Kharazishvili | first2 = A.
 | doi = 10.1023/A:1022102312024
 | issue = 3
 | journal = Georgian Mathematical Journal
 | mr = 1679442
 | pages = 201–212
 | title = On uncountable unions and intersections of measurable sets
 | volume = 6
 | year = 1999| doi-broken-date = 17 March 2025
 }}.</ref>