Hidden structure in the randomness of the prime number sequence?

We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes, that would eventually appear in applications. Finally, our theory allows us to link with three different but important topics: the Hardy-Littlewood conjecture, the statistical mechanics of spin systems, and the celebrated Sierpinski fractal.Comment: 13 pages, 5 figures. New section establishing connection with the Hardy-Littlewood theory. Published in the journal where the solved problem was first describe

On the Diophantine properties of lambda-expansions

For $\lambda \in (1/2, 1)$ and $\alpha$, we consider sets of numbers $x$ such that for infinitely many $n$, $x$ is $2^{-\alpha n}$-close to some $\sum_{i=1}^n \omega_i \lambda^i$, where $\omega_i \in \{0,1\}$. These sets are in Falconer's intersection classes for Hausdorff dimension $s$ for some $s$ such that $- \frac{1}{\alpha} \frac{\log \lambda}{\log 2} \leq s \leq \frac{1}{\alpha}$. We show that for almost all $\lambda \in (1/2, 2/3)$, the upper bound of $s$ is optimal, but for a countable infinity of values of $\lambda$ the lower bound is the best possible result.Comment: 21 page