For an integer $d\ge 3$, put $\Delta_d=\min{2^{d-1},d(d-1)}$. Let $a/q$ be reduced, let $P(X)=\frac{a}{q}X^d+\alpha_{d-1}X^{d-1}+\cdots+\alpha_0$, and let $\mathcal{I}$ be an interval of $H\le q$ consecutive integers. We prove $\left|\sum_{n\in\mathcal{I}}e(P(n))\right|\ll_{d,\varepsilon}q^{1/d}H^\varepsilon+H^{1-1/\Delta_d+\varepsilon}$. Consequently, for every prime $p>d$, every degree-$d$ polynomial $P\in\mathbb{F}p[X]$, and every interval $\mathcal{I}$ of $H$ consecutive integers with $p^{1/d}<H<p^{1/(d-1)}$, writing $H^d/p=H^u$, one has $\left|\sum{n\in\mathcal{I}}e_p(P(n))\right|\ll_{d,\varepsilon}H^{1-\min{u/d,1/\Delta_d}+\varepsilon}$. This strictly improves throughout the full natural short-interval window the best generic estimate obtained by combining classical Weyl differencing with the optimal Vinogradov mean value theorem.
Let $d\geq 4$ and let $R>0$. When $d=4$, assume that $R^2\in\mathbb{N}\setminus 4\mathbb{N}$; when $d\geq 5$, let $R^2\in\mathbb{N}$ be arbitrary. We prove the fixed-radius estimate $$\|A_R f\|_{\ell^{p'}(\mathbb{Z}^d)}\leq C_{d,p,\varepsilon}R^{-d(2/p-1)+\varepsilon}\|f\|_{\ell^p(\mathbb{Z}^d)}$$ for $(d+2)/d\leq p\le...
Let $d,K,N\in \mathbb{N}$ with $K\geq 3$ and $d\geq 4K+4$. Let $\Delta\subset \mathbb{Z}^d$ be the vertex set of a nondegenerate $(K-1)$-simplex, and let $A\subseteq[N]^d$ contain no nontrivial similar copy of $\Delta$. We prove that \[ |A|\ll_{\Delta,d} N^d\exp\!\left(-c_{\Delta,d}\sqrt{\log N}\right) \] improving upo...
Andrew Lott, Á. Magyar, N. R. Ponagandla· 0 citations
Let $p$ be a prime, $d\ge 2$, $H\in [1,p)$, and $\ln\ln p = o(\ln H)$. We prove that $$ \min_{1\le n \le H} n \left\|\frac{a_1 n}{p}\right\|\ldots \left\|\frac{a_d n}{p}\right\| \approx \frac{1}{(\ln p)^{d-1} \ln H} $$ for"almost all"$a \in ({\Bbb Z} / p{\Bbb Z})^d$.
In this paper, we consider the asymptotic density of $T_c(x):=\#\{p\leq x:P^+(p-1)\geq p^c\}$, where $P^+(n)$ denote the largest prime factor of $n$. We show that for $x\rightarrow\infty$, for any $0<c\leq 1/2$, one has \begin{align*} \mathop{\lim\inf}_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}&\geq\max\left(1-\frac{16}...
Let $q_m=2m/(m+1)$ and put \[ \beta_0=\frac{3}{2}+\frac{1}{\log 2}=2.9426950408\ldots, \] where $\log$ is the natural logarithm. We give a proof scheme showing that, for every $\varepsilon>0$, there is $C_\varepsilon<\infty$ such that every complex-valued function $f:\{-1,1\}^n\to\C$ of Fourier degree at most $m$ satis...
Joseph Slote, Chun-Kai Tseng, Alexander Volberg· 0 citations
Let $(a_n)_{n\geqslant 0}$ be a sequence of integers. Its dual sequence $(a_n^*)_{n\geqslant 0}$ is defined by \begin{equation*} a_n^* := \sum_{k=0}^{n} \binom{n}{k}(-1)^k a_k. \end{equation*} Let $p>3$ be a prime. In this paper we mainly investigate congruences modulo $p^2$ involving central binomial coefficients and...
Y. Otmani· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
