An exciting recent development in Diophantine approximation is the paper (Bénard, He, and Zhang 2026), which has just been accepted to the Journal of the American Mathematical Society (JAMS). In this work, Bénard, He, and Zhang make a major advance in our understanding of how well irrational numbers can be approximated by rational ones, even if one restricts attention to irrational numbers from fractal sets such as the Cantor set.
One of the earliest theorems in mathematics asserts that the diagonal of a square of side length cannot be measured exactly using any unit obtained by subdividing the centimetre into equal parts. In modern language, this is the statement that is irrational. Since then, a central question in number theory has been to quantify how well an irrational number can be approximated by rational numbers . We may (and will) always assume that .
There is an inherent trade-off between the quality of the approximation and the size of the denominator . A classical result from the theory of continued fractions, which is also a corollary of Dirichlet’s famous approximation theorem, states that for any irrational the inequality is satisfied for infinitely many pairs of integers and .
In 1891, Hurwitz sharpened this result. He proved that for and every there exist infinitely many integers such that Hurwitz’s theorem is optimal: for the golden ratio and for any , inequality has only finitely many solutions in integers .
A natural refinement of this question is to ask what level of approximation is achieved for a “typical” real number, rather than for all real numbers. Let , where . We say that a real number is -approximable if the inequality has infinitely many solutions in integers . Denote by the set of all -approximable real numbers.
In a landmark 1924 paper, Khintchine (Khintchine 1924) proved the following fundamental theorem.
If and the sequence is non-increasing, then Lebesgue-almost every belongs to .
Conversely, if , then Lebesgue-almost every does not belong to .
For instance, applying part (i) with shows that for any and for almost all , inequality has infinitely many solutions in integers .
In 1984, Mahler (Mahler 1984) posed a striking question: how well can irrational points in Cantor’s middle-third set be approximated by rational numbers? Recall that the Cantor set consists of those real numbers whose base- expansions omit the digit . Since has Lebesgue measure zero, Khintchine’s original theorem does not apply in this setting.
The recent paper (Bénard, He, and Zhang 2026) not only resolves Mahler’s question completely, but does so in a far more general framework.
Recall that the Borel -algebra is the smallest -algebra containing all open subsets of . A Borel probability measure on is a measure defined on with total mass . Such a measure is called self-similar if it satisfies where , the weights are positive and sum to , and the maps are invertible affine contractions with no common fixed point.
As an immediate consequence, consider the self-similar measure supported on the Cantor set , obtained by taking , , and Theorem then implies that almost every is -approximable if , and that almost no is -approximable if the series converges. This provides a complete and definitive answer to Mahler’s question.