Related papers: Remarks on a Problem of Eisenstein

We consider a generalization of the quaternion ring $\Big(\frac{a,b}{R}\Big)$ over a commutative unital ring $R$ that includes the case when $a$ and $b$ are not units of $R$. In this paper, we focus on the case $R=\mathbb{Z}/n\mathbb{Z}$…

For an irrational number $\alpha\in\mathbb{R}$ we consider its irrationality measure function $$ \psi_\alpha(t) = \min_{1\le q\le t,\, q\in\mathbb{Z}} \| q\alpha \|. $$ Let $\boldsymbol{\alpha} = (\alpha_1, \dots, \alpha_n)$ be $n$-tuple of…

For real $\xi$ we consider irrationality measure function $\psi_\xi (t) = \min_{1\le q \le t, \, q\in \mathbb{Z}} ||q\xi||$. We prove that in the case $\alpha \pm \beta \not\in \mathbb{Z}$ there exist arbitrary large values of $t$ with…

We prove that there exists a rearrangement of the first $N$ elements of the trigonometric system such that the $L^2$-norm of the square variation operator is at most $O_{\epsilon}(\log^{9/22+\epsilon}(N))$. This is an improvement over…

We prove that when n>=5, the Dehn function of SL(n,Z) is at most quartic. The proof involves decomposing a disc in SL(n,R)/SO(n) into a quadratic number of loops in generalized Siegel sets. By mapping these loops into SL(n,Z) and replacing…

Let $K/\mathbb{Q}$ be a quadratic extension. In this paper we study the $4$-rank of the class group $\text{Cl}(K(\sqrt{n}))$, where $n$ varies over squarefree rational integers. We show that for $100\%$ of squarefree $n$, the $4$-rank is…

Eberhard and Pohoata conjectured that every $3$-cube-free subset of $[N]$ has size less than $2N/3+o(N)$. In this paper we show that if we replace $[N]$ with $\mathbb{Z}_N$ the upper bound of $2N/3$ holds, and the bound is tight when $N$ is…

In this paper, we prove a result on the $2$-adic logarithm of the fundamental unit of the field $\mathbb{Q}(\sqrt[4]{-q}) $, where $q\equiv 3\bmod 4$ is a prime. When $q\equiv 15\bmod 16$, this result confirms a speculation of Coates-Li and…

Let $C$ be a modular category of Frobenius-Perron dimension $dq^n$, where $q$ is a prime number and $d$ is a square-free integer. We show that if $q>2$ then $C$ is integral and nilpotent. In particular, $C$ is group-theoretical. In the…

Quantum systems with a finite number of states at all times have been a primary element of many physical models in nuclear and elementary particle physics, as well as in condensed matter physics. Today, however, due to a practical demand in…

Suppose that $A \subset \{1,\dots, N\}$ has no two elements differing by a square. Then $|A| \ll N e^{-c\sqrt{\log N}}$.

We prove that there are at least $1.284 \cdot \sqrt{s/\log s}$ irrational numbers among $\zeta(3)$, $\zeta(5)$, $\zeta(7)$, $\ldots$, $\zeta(s-1)$ for any sufficiently large even integer $s$. This result improves upon the previous finding…

Let $p/q$ ($p, q \in \mathbb{N}^*$) be a positive rational number such that $p > q^2$. We show that for any $\epsilon > 0$, there exists a set $A(\epsilon) \subset [0, 1[$, with finite border and with Lebesgue measure $< \epsilon$, for…

We propose a notion of Frobenius-Perron dimension for certain free $\mathbb{Z}$-modules of infinite rank and compute it for the $\mathbb{Z}$-modules of finite dimensional complex representations of unitary groups with nonnegative dominant…

It is proved that Epstein's zeta-function $\zeta_{Q}(s)$, related to a positive definite integral binary quadratic form, has a zero $1/2 + i\gamma$ with $ T \leq \gamma \leq T + T^{{3/7} +\varepsilon} $ for sufficiently large positive…

We discover a non-trivial relation between the mock modular generating functions of the level $1$ and level $N$ Hurwitz class numbers. This relation yields a holomorphic modular form of weight $\frac{3}{2}$ and level $4N$, where $N > 1$ is…

In this paper we compute the gonality over Q of the modular curve X1(N) for all N <= 40 and give upper bounds for each N <= 250. This allows us to determine all N for which X1(N) has infinitely points of degree <= 8. We conjecture that the…

We show that all perfect odd integer squares not divisible by 3, can be usefully written as sqrt(N) = a + 18p, where the constant a is determined by the basic properties of N. The equation can be solved deterministically by an efficient…

We provide explicit bounds for the number of integral ideals of norms at most $X$ is $\mathbb{Q}[\sqrt{d}]$ when $d <0$ is a fundamendal discriminant with an error term of size $O(X^{1/3})$. In particular, we prove that, when $\chi$ is the…

The Gelfand--Zetlin basis for representations of $U_q(sl(N))$ is improved to fit better the case when $q$ is a root of unity. The usual $q$-deformed representations, as well as the nilpotent, periodic (cyclic), semi-periodic (semi-cyclic)…