22 problems

Minimum-dimension conjecture. The infimum is attained at $\beta_3$.

Full dimension conjecture. For all $\lambda$ sufficiently close to $1$, one should have

Benjamini–Solomyak pair-correlation conjecture. For almost every $\lambda\in(1/2,1)$, there exist $c,C>0$ such that

For $\lambda$ in the open unit disk, let … be the associated self-similar set, and let $\nu_\lambda$ be the corresponding complex Bernoulli convolution. In the case…

Varjú's conjecture. For $\lambda>1/2$, the condition that $\lambda^{-1}$ is not a Pisot number should characterize the absolute continuity of $\mu_\lambda$.

Tribonacci parameter conjecture. The minimizer is the tribonacci parameter:

Let $D_2(\mu_\lambda)$ be the correlation dimension of the Bernoulli convolution measure $\mu_\lambda$, and let $\lambda$ be either the reciprocal of the Fibonacci parameter … or t…

Let $\epsilon>0$, and let $I_{\beta_n}$ denote the interval associated with the Pisot number $\beta_n$, $\mu_{I_{\beta_n}}$ the corresponding measure, and…

Given a number $\beta\in(1,2)$, let $\nu_{\beta}$ denote the associated Bernoulli convolution and let $\dim_H$ denote Hausdorff dimension. Large-degree Pisot dimension conjecture.…

Let $\nu_{\beta_1}$ denote the Bernoulli convolution associated with $\beta_1$. Suppose that $\beta_1\in(1,2)$, $\beta_1\neq(1+\sqrt{5})/2$, and that $\beta_1$ has a real conjugate…

Ngai–Wang conjecture. The measure $\nu_{\lambda_1,\lambda_2}^{1/2}$ is absolutely continuous for almost all $\lambda_1,\lambda_2\in(0,1)$ such that $\lambda_1\lambda_2>1/4$.

For $<\lambda<1$, let $\nu_\lambda$ denote the corresponding Bernoulli convolution, namely the distribution of the random series … where the $\epsilon_n$ are independent random sig…

Local-dimension conjecture. With the exception of weak Perron parameters $\beta$, the only value of the local dimension larger than one for a Bernoulli convolution is

Let $\nu_t$ denote the measure associated with the parameter $t$, and call an intersection of periodic address curves supercritical when it has the supercritical property described…

Let $A_\lambda$ be the support of the complex Bernoulli convolution associated to a non-zero complex number $\lambda$ in the open unit disk $\mathbb{D}$, and let … be its connected…

Let $\mu_p$ and $\mu_q$ be the self-similar measures associated with the iterated function system under consideration, and let $W_1(\mu_p,\mu_q)$ denote their first Wasserstein dis…

Counting beta-expansions conjecture. For every such $\beta$, the displayed formula holds for almost all $x\in I_{\beta}$.

Let $\nu_{\beta}$ be the Bernoulli convolution associated with $\beta$, and let $f_n$ denote the sequence of approximating functions introduced in the paper. Convergence conjecture…

For $\lambda \in \left(\frac{1}{2},1\right)$, let the infinite Bernoulli convolution be the probability distribution of the random variable … where the signs are chosen independent…

Let $\sigma=\frac{1}{2}(D_0+D_1)$ and let $\mu$ be the invariant measure of the associated infinite Bernoulli convolution with contraction ratio $\delta\in(1/2,1)$. The measure…

Orthonormal exponential conjecture. There are orthonormal complex exponentials for $\nu_{\lambda}$ if and only if $\lambda=1/(2n)$.

Linear pair-correlation conjecture. For almost every $\lambda\in(\frac{1}{2},1)$, there exist $c,C>0$ such that