Fix $k\ge 3$, and let $f_k(N)$ be the largest harmonic sum of a subset of $[N]$ containing no $k$ distinct integers with a common pairwise least common multiple. We prove that $f_k(N)=(\log N)^{\gamma_k+o(1)}$ for a well-defined exponent $\gamma_k\in(0,1]$. Following the weighted-pressure idea of Chojecki, we give a self-contained proof of variational formulas for $\gamma_k$ in terms of weighted sunflower-free families. We then eliminate the continuous weight: if $M_k(n,r)$ is the largest size of an $r$-uniform $k$-cosunflower-free family on $[n]$, then \[ \gamma_k=\sup_{n\ge1,\ 1\le r\le n}\frac{r}{en}M_k(n,r)^{1/r}. \] This finite-block formula yields a direct transfer principle from uniform set-system constructions, recovers the Tang--Zhang bounds, and gives $0.438899\ldots<\gamma_3\le0.889881\ldots$.
Let $S_k(\alpha;K)$ denote the exponential sum over $k$-free integers in the short interval $(N-K,N]$. For $s>0$, we prove essentially tight bounds on the $s$-th moments of $S_k(\alpha;K)$ whenever $K \gg N^{\theta_{k,s}+\epsilon}$ for some $\theta_{k,s}<1/2$. As an immediate consequence, we obtain a lower bound for th...
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 th...
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...
Let $S^k_d(n)$ denote the maximum number of regular $(k-1)$-simplices spanned by $n$ points in $\mathbb{R}^d$. For all fixed $r\geq k\geq3$, the exact value of $S^k_{2r}(n)$ was recently determined by Dumitrescu and the authors for all sufficiently large $n$ when $k=3$, and conditionally on an optimization problem when...
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...
Let $F$ be a field, let $0\le \ell\le k-2$, and suppose that $n\ge k-1$. We determine the minimum degree of a polynomial in $F[x_1,\ldots,x_n]$ that vanishes to order at least $k$ at every nonzero vertex of the Boolean cube and to order exactly $\ell$ at the origin. The answer is \[ n+2k-2-\rho_F(k-\ell), \] where $\rh...
D. Menezes· 0 citations
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