On points contain arithmetic progressions in their lüroth expansion

For any x ∈ (0, 1] (except at most countably many points), there exists a unique sequence {dn(x)}n≤1 of integers, called the digit sequence of x, such that The dexter infinite series expansion is called the Lüroth expansion of x. This paper is concerned with the size of the set of points x whose digit sequence in its Lüroth expansion is strictly increasing and contains arbitrarily long arithmetic progressions with arbitrary common difference. More precisely, we determine the Hausdorff dimension of the above set.