On the theorem of Jarník and Besicovitch

Arithmetic Progressions in Sets of Fractional Dimension

“…Kaufman's set [17], unlike those of Salem and Kahane, is completely deterministic. It is easily seen that if x belongs to Kaufman's set, then so do 2x, 3x, .…”

“…It is quite difficult to construct explicit examples of Salem sets with 0 < dim H (E) < 1. Such constructions are due to Salem [25], Kaufman [17], Bluhm [2], [3]; we give an alternative construction in Section 6. On the other hand, Kahane [16] showed that Salem sets are ubiquitous among random sets, in the sense that images of compact sets under Brownian motion are almost surely Salem sets.…”

Salem sets in local fields, the Fourier restriction phenomenon and the Hausdorff–Young inequality

“…Salem's construction of such sets was probabilistic. Deterministic constructions of Salem sets on the real line are rare; however see [8] and [6]. We follow Salem's probabilistic approach to proving the existence of generalized Cantor type Salem sets in an ultrametric local field K. The existence of the norm | · | and the local compactness property of K allow us to adjust all the above real analysis to the local field setting.…”

T‐numbers form anM0set

“…A rather different measure of the size of sets of real numbers arises in harmonic analysis. In [6] Kaufman showed that badly approximable numbers carry a measure in M o and in [7] he has shown that well approximable numbers also carry a measure in M o . This is a subtle distributional property which is not characterized by, for example, merely the Lebesgue measure of E (cf.…”