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 the $L^1$-mean of the M\"obius-twisted exponential sum over short intervals of length at least $N^{0.49685}$. Moreover, we show that further improvements on all of these results would follow immediately from improvements to an $\ell^2$-estimate involving the M\"obius function.
Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1...
Fix integers $0\leq k<n$. Let $F_{n,k}(N)$ be the least size of a finite set $B\subset\Z^n$ that contains a filled axis-parallel cube $k$-skeleton centered at each point of some $N$-point set. We prove that $F_{n,k}(N)$ has order $N^{1-(n-k)/(2n^2)}$, with constants depending only on $n$ and $k$. Thornton proved the up...
We prove that for every fixed $\lambda>0$ and all sufficiently large $n$, any $z_1,\dots,z_n\in\C$ with $|z_j|\geq1$ satisfy $\max_{2\leq k\leq n+1}|\sum_j z_j^k|>e^{-\lambda n}$. Consequently, the $n$th root of the optimal maximum tends to $1$, so no constant $C>1$ in Erd\H{o}s 973 can exist. The proof combines a trun...
For a simple graph $G$ of order $n$, let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ denote its adjacency eigenvalues. Hong's problem asks for the optimal upper bound for $\lambda_k(G)$. A recent theorem of Sivashankar gives, for every $k\ge3$, \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1, \] with sharp exam...
Hitesh Kumar, Bojan Mohar, S. A. Mojallal et al.· 0 citations
For a convex body $K \subset \mathbb R^d$ let $\Delta(K)$ be the expected distance between two independent uniform points of $K$, and let $\theta(K)$ be the corresponding expectation for normalized surface measure on $\partial K$. The Zaporozhets-Tarasov conjecture asserts $\Delta(K) \le \theta(K)$. We prove this conje...
In this paper, we refine Borel's 1903 result in Diophantine approximation by providing sharper bounds for the minimum of three consecutive approximation coefficients $\Theta_n(x)$, defined for any real number $x$ with regular continued fraction (RCF) expansion $x=[0;a_1,a_2,\dots]$ as $\Theta_n = q_n^2\left| x-\frac{p_...
Savita S. Adhin, Ayreena Bakhtawar, C. Kraaikamp· 0 citations
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