arXiv subjects
Publications and source records attributed to Katerina Saettone.
This paper presents some new results concerned with uniform distribution properties associated with the sequence $(a_n)_{n\in\mathbb{N}}$, which is defined as the distance from the $n$-th square pyramidal number to the closest square. We also extend the results to arithmetic progressions.
On the occasion of George Andrews' and Bruce Berndt's combined 170th birthday, we prove two conjectures of Sir Frederick Pollock that are more than 170 years old. In 1843, Pollock conjectured that every positive integer is the sum of at most 13 icosahedral numbers, or at most 21 dodecahedral numbers. We establish refined versions of both conjectures.
We generalize the Cannonball Problem by introducing integer-valued and non-increasing arithmetic functions $w$. We associate these functions $w$ with certain polygons, which we call cannonball polygons. Through this correspondence, we show that for any $Z\in\mathbb{N}$, there exists a cannonball polygon with multiplicity 8 and largest side of length $Z$. Moreover, for any multiplicity $s$ greater than 8, we provide an asymptotic formula for the number of distinct classes of cannonball polygons with multiplicity $s$.
We provide an asymptotic formula for the average value of the sequence A351830: $a_{n} = |P_{n} - y^{2}_{n}|$ for $1 \leq n \leq x$, where $P_{n}$ is the $n$-th square pyramidal number and $y^{2}_{n}$ is the closest square to $P_{n}$. Moreover, we supply asymptotic formulas for the $k$-th moment of the same sequence, for any fixed natural number $k$.
Let $p$ be an odd prime, and let $ω$ be a primitive $p$th root of unity. In this paper, we introduce a metric on the cyclotomic field $K=\mathbb{Q}(ω)$. We prove that this metric has several remarkable properties, such as invariance under the action of the Galois group. Furthermore, we show that points in the ring of integers $\mathcal{O}_K$ behave in a highly uniform way under this metric. More specifically, we prove that for a certain hypercube in $\mathcal{O}_K$ centered at the origin, almost all pairs of points in the cube are almost equi-distanced from each other, when $p$ and $N$ are large enough. When suitably normalized, this distance is exactly $1/\sqrt{6}$.