In 1943, Erd\H{o}s considered the minimum number $f(n)$ of terms between two fractions in the Farey sequence of order $n$ whose numerators and denominators are oppositely ordered. Determining the constant $c$ in $f(n)=(c+o(1))n$ is known as Erd\H{o}s Problem 1005. Recently, Cipollini solved this asymptotic problem by proving that $f(n)=(1/4+o(1))n$. Following his framework, we give an analytic proof of an exact formula for $f(n)$ for all sufficiently large $n$. Combining this with a finite computer verification, we further determine $f(n)$ for every integer $n\geq 4$.
Following Erd\H{o}s (1982) and Sanna (2019), we study the arithmetic function $h(n)$, which is defined to be the number of distinct exponents in the prime factorization of a positive integer $n$. Among other things, we show that $$ \sum_{n\leq x}h(\phi(n)) \asymp x\left(\frac{\log\log x}{\log\log\log x}\right)^{1/2}, $...
Mikhail R. Gabdullin, V. V. Iudelevich· 0 citations
For a compact set $K\subset\mathbb{C}$, let $\vartheta(K)$ be the infimum of the planar areas of the unit lemniscates of all monic polynomials with zeros in $K$, allowing arbitrary degree and repeated zeros. We prove that $\vartheta(K)=0$ whenever $\operatorname{cap}(K)=1$, with no regularity assumption on $K$. The pro...
For practical $N$ let $h(N)$ be the least $k$ such that every integer $1\le m\le N$ is a sum of at most $k$ distinct divisors of $N$. We prove $h(n!)\le(2\log2+o(1))\,n/\log n$. This improves the bounds of order $n/(\log n)^{1/2-\varepsilon}$ established in Tenenbaum-Yokota's Lemma 4 and Yokota's 1995 knapsack note. We...
We prove a quantitative neighboring-denominator variant of the Erd\H{o}s-Mahler conjecture. Let $p_n/q_n$ be the convergents of an irrational real number $\xi$. If $p_nq_nq_{n+1}$ is $S$-smooth for infinitely many $n$, where $S$ is a fixed finite set of primes, then there exists an effectively computable constant $c=c(...
We determine the optimal exponential scale in Erd\H{o}s's robust polynomial interpolation problem and identify the leading exponential coefficient for Chebyshev--Lobatto grids. Given $C>0$ and $0\le\rho<1$, put $H=C/(1-\rho)$. We prove that every configuration of $n\ge n_0$ nodes in $[-1,1]$ admits sign data for which...
For a finite set $S$ of positive integers, put $K(S):=-\min_{x\in\mathbb T}\sum_{s\in S}\cos(2\pi sx)$. Bedert recently proved the uniform lower bound $K(S)\geq |S|^{1/5-o(1)}$. We remove the subpolynomial loss and prove that $K(S)\geq c|S|^{1/5}$ for an absolute constant $c>0$. The proof combines two estimates from Be...
Abhishek Shankar· 1 citation· ⚡1
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