We consider the problem of finding lower bounds for integer linear combinations of $\sqrt{a_1},\ldots,\sqrt{a_K}$, where $a_1,\ldots,a_K$ are positive integers such that their square roots are linearly independent over the rationals. We use a probabilistic approach and prove that for any integers $m_1,\ldots,m_K$, not all zero, $$\left|\sum_{n\leq K}m_n\sqrt{a_n}\right|>e^{-1/2}\left(\max_{n\leq K}|m_n|\sqrt{a_n}\cdot\sqrt{K}\right)^{-(2^{K-1}-1)}.$$ This inequality improves the dependence on $K$ in the classical product bound.
Let $A=(A_n)_{n\ge2}$ be a triangular array of random matrices, where $A_n=(a_{ij})_{1\le i,j\le n}$ is an $n\times n$ random matrix with independent real entries satisfying $\mathbb E a_{ij}=0$ and $\mathbb Ea_{ij}^2=1$, and put $\mathcal L_n=\log|\det A_n|$ and \[ W_n^{\mathrm d}(A_n):=\frac{\mathcal L_n - \frac12\lo...
For a positive integer $t$, let $d_t$ denote the natural density of the set of $n$ for which $t$ is a sum of distinct divisors of $n$. Erd\H{o}s proved that $d_t$ exists, gave an unspecified polylogarithmic upper bound, asserted without proof a matching lower bound, and asked whether $d_t \sim c_3/(\log t)^{c_4}$. We r...
Let $d \geq 1$ be fixed and let \[ D(n,d) := \sum_{j=0}^{d} \binom{n-1}{j}. \] We show that if $x^{(1)}, \dots, x^{(m)}$ are independent uniform points of $\{\pm 1\}^n$ then uniformly for $m \leq D(n,d)$, there exists a constant $C_d>0$ such that \[ \mathbb{P}((x^{(1)})^{\otimes d}, \dots, (x^{(m)})^{\otimes d} \text{...
For a fixed integer $n\ge 2,$ we consider the homogeneous polynomial $$ P(x_1, x_2, \ldots, x_{n+2})=\sum_{i=1}^{n} (x_2-x_1)^{i-1} x_1^{n-i} x_{i+2}. $$ We prove that, for any finite set $A$ of complex numbers, $$ \Bigl|\bigl\{P(x_1,x_2,\ldots,x_{n+2}): \, x_i\in A\bigr\}\Bigr|\gg |A|^{n}. $$ The implicit constant in...
For $n$-dimensional simultaneous best Diophantine approximations in an arbitrary norm induced by an inner product, for $n\geq2$ we prove $q_{k+2^n}\geq q_k+\min\{q_{k+2^{n-1}},2q_{k+1}\}$. This yields $g_n(\alpha):=\liminf_{m\to\infty}(q_m)^{1/m}\geq\varphi^{1/2^{n-1}}, \text{ for } \ \varphi = \dfrac{1 + \sqrt{5}}{2}....
In this note, we study the infinite reciprocal sum $\sum_{k=n}^{\infty}1/B_k^4$ involving the fourth powers of balancing numbers $B_n$. We show that, for every $n\geq2$, \begin{equation*} \left\lfloor \left( \sum_{k=n}^{\infty}\frac{1}{B_k^4} \right)^{-1} \right\rfloor = B_n^4-B_{n-1}^4 -\left\lceil\frac{B_{2n-1}}{280}...
S. Panda, Aditya Kumar Dash, U. K. Dutta· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
