Let \(\mathbf{S}=\{S_k\}\) be a sequence of measurable subsets of \(\mathbb{R}\) and \(\mu\) be a probability measure in \(\mathbb{R}\) . The maximal operators, associated to \(\mathbf{S}\) and \(\mu\) respectively, are defined as follows \[ M_\mathbf{S}(f)(x) =\sup_{r>0,k\in \mathbb{N} }\frac{1}{|S_k|}\int_{S_k}|f(x+ry)|dy, \]
\[ M_\mu(f)(x) =\sup_{r>0}\int|f(x+ry)|d\mu(y). \] For any \(\varepsilon \in [0,1/3)\) the authors prove the existence of a decreasing sequence of sets \(\mathbf{S}=\{S_k\}\) with the properties:

(a)

\(S_k\subset[1,2]\).

(b)

Each \(S_k\) is a disjoint union of finitely many intervals.

(c)

\(|S_k|\rightarrow 0 (k\rightarrow\infty)\).

(d)

The weak-\(\ast\) limit \(\mu\) of the densities \(\mathbf{1}_{S_k}/|S_k|\) exists.

(e)

\(\bigcap_{k=1}^{\infty}S_k \) has Hausdorff dimension \(1-\varepsilon\).

(f)

The maximal operators \(M_\mathbf{S}\) and \(M_\mu\) are bounded on \(L^{p}(\mathbb{R})\) for any \(p>\frac{1+\varepsilon}{1-\varepsilon}\).

From this result, as a corollary, a differentiation theorem is obtained for averages on \(rS_k\) and with respect to \(\mu\): for every \(f\in L^{p}(\mathbb{R})\) with \(p>\frac{1+\varepsilon}{1-\varepsilon}\), \[ \lim_{r\rightarrow 0} \sup\limits_{k}\big| \frac{1}{r|S_k|}\int_{x+r S_k}f(y)dy-f(x)\big|=0 \] and \[ \lim_{r\rightarrow 0} \big| \int f(x+ry)d\mu(y) -f(x)\big|=0 \] for a.e. \(x\in\mathbb{R}\).
The last result gives a positive answer to a question of Aversa and Preiss on differentiation theorems for averages over sparse one dimensional sets.
Some other generalizations and corollaries are also given.
Complicated proofs combine probabilistic techniques with the methods developed in multidimensional harmonic analysis.