We present a complete characterization of two non-integers with the same Rényi-Parry measure. We prove that for two non-integers $\beta_1 ,\beta_2 >1$, the Rényi-Parry measures coincide if and only if $\beta_1$ is the root of equation $x^2-qx-p=0$, where $p,q\in\mathbb{N}$ with $p\leq q$, and $\beta_2 = \beta_1 + 1$, which confirms a conjecture of Bertrand-Mathis in [A. Bertrand-Mathis, Acta Math. Hungar. 78, no. 1-2 (1998):71–78].

Integers obeying a digital restriction, such as having no 7s in their base 10 representation, are a discrete analog of the Cantor set and have been a recent topic of interest in analytic number theory. Smooth integers, which are integers having only small prime factors, are an important class of integers with applications to many different areas of math. In this talk, we find an asymptotic on the intersection between the two.

Let $K$ be the ternary Cantor set, and let $\mu$ be the Cantor–Lebesgue measure on $K$. It is well known that every point in $K$ is not 3-normal. However, if we take any natural number $p \ge 2$ that is not a power of 3, then $\mu$-almost every point in $K$ is $p$-normal. This classical result is due to Cassels and W. Schmidt.

Another way to obtain normal numbers from K is by rescaling and translating $K$, then examining the transformed set. A recent nice result by Dayan, Ganguly, and Barak Weiss shows that for any irrational number $t$, for $\mu$-almost all $x \in K$, the product $tx$ is 3-normal.

In this talk, we will discuss these results and their generalizations, including replacing $p$ with an arbitrary beta number and considering more general times-3 invariant measures instead of the Cantor–Lebesgue measure.

The classical Catalan and Motzkin numbers have remarkable continued fraction expansions, the corresponding sequences of Hankel determinants consist of -1, 0 and 1 only. We find an infinite family of power series corresponding to q-deformed real numbers that have very similar properties. Moreover, their sequences of Hankel determinants turn out to satisfy Somos and Gale-Robinson recurrences. (Partially based on a joint work with Emmanuel Pedon.)

Consider the classical middle third Cantor set. This is a self-similar set containing all the numbers in the unit interval which have a ternary expansion that avoids the digit 1. We can ask when the intersection of the Cantor set with a translate of itself is also self-similar. Sufficient and necessary conditions were given by Deng, He, and Wen in 2008. This question has also been generalized to classes of subsets of the unit interval. I plan to discuss how existing ideas can be used to address the question for certain self-similar sets with dimension greater than one. These ideas will be illustrated using a class of self-similar sets in the plane that can be realized as radix expansions in base $-n+i$ where $n$ is a positive integer. I will also discuss a property of the fractal dimensions of these kinds of intersections.

In the realm of integer factorization, certain methods, such as CFRAC, leverage the properties of continued fractions, while others, like SQUFOF, combine these properties with the tools provided by quadratic forms. Recently, Michele Elia revisited the fundamental concepts of SQUFOF, including reduced quadratic forms, distance between quadratic forms, and Gauss composition, offering a new perspective for designing factorization methods.

In this seminar, we present our algorithm, which is a refinement of Elia's method, along with a precise analysis of its computational cost. Our algorithm is polynomial-time, provided knowledge of a (not too large) multiple of the regulator of $\mathbb{Q}(\sqrt{N})$. The computation of the regulator governs the total computational cost, which is subexponential, and in particular $O(\exp(\frac{3}{\sqrt{8}}\sqrt{\ln N \ln \ln N}))$. This makes our method more efficient than CFRAC and SQUFOF, though less efficient than the General Number Field Sieve.

We identify a broad family of integers to which our method is applicable including certain classes of RSA moduli. Finally, we introduce some promising avenues for refining our method. These span several areas, ranging from Algebraic Number Theory, particularly for estimating the size of the regulator of $\mathbb{Q}(\sqrt{N})$, to Analytic Number Theory, particularly for computing a specific class of $L$-functions.

Joint work with Nadir Murru.

The goal is to explore the recently found interplay between integer partitions and the dynamics of the triangle map (a type of multi-dimensional continued fraction algorithm). (This is joint work with Baalbaki, Bonanno, Del Vigna and Isola.)

As we will see, the triangle map gives an almost internal symmetry from the set of integer partitions to itself, which in turn allows the generation of any number of new partition identities.

Further, this allows us to place a tree structure on the space of all integer partitions. (This is joint work with Joe Fox and with Jacob Lehmann Duke). This tree structure allowed us to find the natural extension of the triangle map in any dimension. As with the classical Farey map, the dynamics of this map, in every dimension, has an indifferent fixed point, which in turn can be used to understand the structure of the integer partition tree.

Among the many different types of multi-dimensional continued fractions that exist, for still unknown reasons it appears that the triangle map is the only one that is “partition” compatible.

Thus we use the triangle map (stemming from number theory and dynamics) to understand classical integer partition numbers from combinatorics, and use partition numbers to understand the dynamics of the triangle map.

Kolam is a form of traditional decorative art in India, that is drawn by using rice flour, white stone powder, chalk or chalk powder. It is often practised by women in front of their house entrance. One particular kolam uses a 4×4 grid. It can also be found in the Vanuatu Islands, in Africa, etc. We show that a natural generalization on grids of size 8×8, 16×16, etc. is linked to… the Thue-Morse sequence. Further we unveil two (twin) morphisms that generate this family of kolam, and show that they appear in unrelated and somewhat unexpected fields. Time permitting we will allude to a vast, relatively new, field: ethnomathematic(s).