We continue our investigation of the fractal uncertainty principle (FUP) for random fractal sets. In the prequel Eswarathasan and Han [‘Fractal uncertainty principle for discrete Cantor sets with random alphabet’, Math. Res. Lett. 30(6) (2023), 1657–1679], we considered the Cantor sets in the discrete setting with alphabets randomly chosen from a base of digits so the dimension $\delta \in (0,\frac 23)$. We proved that, with overwhelming probability, the FUP with an exponent $\ge \frac 12-\frac 34\delta -\varepsilon $ holds for these discrete Cantor sets with random alphabets. In this paper, we construct random Cantor sets with dimension $\delta \in (0,\frac 23)$ in $\mathbb {R}$ via a different random procedure from the previous one used in Eswarathasan and Han [‘Fractal uncertainty principle for discrete Cantor sets with random alphabet’, Math. Res. Lett. 30(6) (2023), 1657–1679]. We prove that, with overwhelming probability, the FUP with an exponent $\ge \frac 12-\frac 34\delta -\varepsilon $ holds. The proof follows from establishing a Fourier decay estimate of the corresponding random Cantor measures, which is in turn based on a concentration of measure phenomenon in an appropriate probability space for the random Cantor sets.

We strengthen known results on Diophantine approximation with restricted denominators by presenting a new quantitative Schmidt-type theorem that applies to denominators growing much more slowly than in previous works. In particular, we can handle sequences of denominators with polynomial growth and Rajchmann measures exhibiting arbitrary slow decay, allowing several applications. For instance, our results yield non-trivial lower bounds on the Hausdorff dimensions of intersections of two sets of inhomogeneously well-approximable numbers (each with restricted denominators) and enable the construction of Salem subsets of well-approximable numbers of arbitrary Hausdorff dimension.

Suppose that both you and your friend toss an unfair coin n times, for which the probability of heads is equal to α. What is the probability that you obtain at least d more heads than your friend if you make r additional tosses? We obtain asymptotic and monotonicity/convexity properties for this competing probability as a function of n, and demonstrate surprising phase transition phenomenon as the parameters d, r, and α vary. Our main tools are integral representations based on Fourier analysis.

Let {λj}j≥0 be a sequence of positive integers such that λj+1/λj≥3 and {aj}j≥0 a sequence of complex numbers such that |aj|≤1. Let μ be the Riesz product πj≥0[1+ Re(ajeiλjx)], that is, the weak limit of measures on T the density of which are the partial products. Then if Σj≥0|aj|2≤∞ the series Σj≥0 aj(eiλjx - ½āj) converges for μ-almost every x. The μ-a.e. convergence of series Σ ajeinλjx is also investigated as well as the case of Riesz products on a compact commutative group.

Assume that we want to estimate – σ, the abscissa of convergence of the Laplace transform. We show that no non-parametric estimator of σ can converge at a faster rate than (log n)–1, where n is the sample size. An optimal convergence rate is achieved by an estimator of the form where xn = O(log n) and is the mean of the sample values overshooting xn. Under further parametric restrictions this (log n)–1 phenomenon is also illustrated by a weak convergence result.