For $k\geq2$, let $\alpha_k$ be the supremum of the exponents $a$ for which almost every integer $n$ has $k$ distinct divisors in a multiplicative interval of relative length $(\log n)^{-a}$. Select each positive integer $i$ independently with probability $1/i$, forming a random set $\mathbf A$, and let $\beta_k$ be the supremum of the $c<1$ for which, with probability tending to one as $D\to\infty$, the set $\mathbf A\cap[D^c,D]$ has $k$ distinct subsets with the same sum. We prove that $\alpha_k=\beta_k/(1-\beta_k)$, resolving a conjecture of Ford, Green and Koukoulopoulos [Invent. Math. 232 (2023), 1027--1160]. We also prove that their weak and strict entropy thresholds coincide. The proof combines flag refinement and entropy concavity with an upper bound for approximate subset sums that is uniform in arbitrary translations. A model with independent geometric prime exponents then transfers this bound to divisors.
For an integer $k\geq2$, let $\alpha_k$ be the supremum of the real numbers $a$ for which almost every integer $n\geq2$ has divisors $d_1<\cdots<d_k\mid n$ satisfying $d_k\leq d_1\bigl(1+(\log n)^{-a}\bigr).$ Let $\mathcal A\subseteq\N$ be the logarithmic random set in which the events $m\in\mathcal A$ are mutually ind...
A subset $A\subseteq \mathbb N_0$ is $k$-AP covering if there exists a constant $n_0$ such that for every integer $x>n_0$, there exists $d\in\mathbb N_0$ such that $x-d, x-2d,\dots,x-(k-1)d$ are all in $A$. Disproving a conjecture of Kiss, S\'andor, and Yang, we prove that for every integer $k\geq 6$, there exists a co...
Let $b_1,\dots,b_d\ge2$ be fixed integers, and let $k$ be their multiplicative rank. We study representations $n=a+b_1^{u_1}\cdots b_d^{u_d}$, where $a$ belongs to a set $\mathcal{A}$ of positive integers and $u_1,\dots,u_d$ are positive integers, counting distinct tuples of exponents separately. Under density and corr...
We denote by $f_k(\Gamma)$ the largest integer with the property that every subset of a finite abelian group $\Gamma$ of size at least $|\Gamma| - f_k(\Gamma)$ is a $k$-fold sumset. Extending a recent result of Alon and Pham, we prove that $$ f_k(\Gamma) \leq \widetilde{O} \left(n^{(2k-1)/(4k-3)}\right) $$ holds for al...
Denote by $\omega(n)$ the number of distinct prime divisors of the natural number $n$. In 2007, Granville and Soundararajan gave a quite new method to compute the higher moments $\sum_{n\leq x}(\omega(n)-\log\log x)^{k}$, for a wide range of integers $k\geq 2$. In this notes, we shall apply the method for $\omega_{z}(n...
T. Minamide, Haruka Sakai, Y. Tanigawa· 1 citation
For a positive integer $n$, let $S(n)=\{\sigma(d): d\mid n\}$ be ordered by divisibility, and let $a(n)$ be its width. We study how much of the divisor lattice of $n$ survives under the map $d\mapsto \sigma(d)$. For every fixed prime-exponent pattern, the largest possible value of $a(n)$ is the width of a corresponding...
Felix Huber· 0 citations
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