Character Sums and Congruences with n!

We estimate character sums with n!, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime p and obtain new information about the spacings between quadratic nonresidues modulo p. In particular, we show that there exists a positive integer $n\ll p^{1/2+\epsilon}, such that n! is a primitive root modulo p. We also show that every nonzero congruence class a \not \equiv 0 \pmod p can be represented as a product of 7 factorials, a \equiv n_1! ... n_7! \pmod p, where$\max \{n_i | i=1,... 7\}=O(p^{11/12+\epsilon}), and we find the asymptotic formula for the number of such representations. Finally, we show that products of 4 factorials $n_1!n_2!n_3!n_4!, with \max\{n_1, n_2, n_3, n_4\}=O(p^{6/7+\epsilon})$ represent ``almost all''residue classes modulo p, and that products of 3 factorials n_1!n_2!n_3! with \max\{n_1, n_2, n_3\}=O(p^{5/6+\epsilon})$ are uniformly distributed modulo p.Comment: 20 pages. Trans. Amer. Math. Soc. (to appear

On some congruences and exponential sums

Let $\varepsilon>0$ be a fixed small constant, ${\mathbb F}_p$ be the finite field of $p$ elements for prime $p$. We consider additive and multiplicative problems in ${\mathbb F}_p$ that involve intervals and arbitrary sets. Representative examples of our results are as follows. Let ${\mathcal M}$ be an arbitrary subset of ${\mathbb F}_p$. If $\#{\mathcal M} >p^{1/3+\varepsilon}$ and $H\ge p^{2/3}$ or if $\#{\mathcal M} >p^{3/5+\varepsilon}$ and $H\ge p^{3/5+\varepsilon}$ then all, but $O(p^{1-\delta})$ elements of ${\mathbb F}_p$ can be represented in the form $hm$ with $h\in [1, H]$ and $m\in {\mathcal M}$, where $\delta> 0$ depends only on $\varepsilon$. Furthermore, let ${\mathcal X}$ be an arbitrary interval of length $H$ and $s$ be a fixed positive integer. If $$H> p^{17/35+\varepsilon}, \quad \#{\mathcal M} > p^{17/35+\varepsilon}.$$ then the number $T_6(\lambda)$ of solutions of the congruence $$\frac{m_1}{x_1^s}+ \frac{m_2}{x_2^s}+ \frac{m_3}{x_3^s}+\frac{m_4}{x_4^s}+ \frac{m_5}{x_5^s}+\frac{m_6}{x_6^s} \equiv \lambda\mod p, \qquad m_i\in {\mathcal M}, \quad \ x_i \in {\mathcal X}, \quad i =1, \ldots, 6,$$ satisfies $$T_6(\lambda)=\frac{H^6(\#{\mathcal M})^6}{p}\left(1+O(p^{-\delta})\right),$$ where $\delta> 0$ depends only on $s$ and $\varepsilon$

Character sums and products of factorials modulo pp

In this paper, we apply character sum estimates to study products of factorials modulo p

Exponential Sums and Congruences with Factorials

We estimate the number of solutions of certain diagonal congruences involving factorials. We use these results to bound exponential sums with products of two factorials $n!m!$ and also derive asymptotic formulas for the number of solutions of various congruences with factorials. For example, we prove that the products of two factorials $n!m!$ with $\max\{n,m\}<p^{1/2+\epsilon}$ are uniformly distributed modulo $p$, and that any residue class modulo $p$ is representable in the form $m!n!+n_1! + ... +n_{49}!$ with $\max \{m,n, n_1, >..., n_{49}\} < p^{8775/8794+ \epsilon}$.Comment: 21 page

On Congruences with Products of Variables from Short Intervals and Applications

We obtain upper bounds on the number of solutions to congruences of the type $$(x_1+s)...(x_{\nu}+s)\equiv (y_1+s)...(y_{\nu}+s)\not\equiv0 \pmod p$$ modulo a prime $p$ with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M. Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. B. Friedlander and H. Iwaniec and some results of M.-C. Chang and A. A. Karatsuba on character sums twisted with the divisor function

Multiplicative Congruences with Variables from Short Intervals

Recently, several bounds have been obtained on the number of solutions to congruences of the type $$(x_1+s)...(x_{\nu}+s)\equiv (y_1+s)...(y_{\nu}+s)\not\equiv0 \pmod p$$ modulo a prime $p$ with variables from some short intervals. Here, for almost all $p$ and all $s$ and also for a fixed $p$ and almost all $s$, we derive stronger bounds. We also use similar ideas to show that for almost all primes, one can always find an element of a large order in any rather short interval

On the Hidden Shifted Power Problem

We consider the problem of recovering a hidden element $s$ of a finite field \F_q of $q$ elements from queries to an oracle that for a given x\in \F_q returns $(x+s)^e$ for a given divisor $e\mid q-1$. We use some techniques from additive combinatorics and analytic number theory that lead to more efficient algorithms than the naive interpolation algorithm, for example, they use substantially fewer queries to the oracle.Comment: Moubariz Garaev (who has now become a co-author) has introduced some new ideas that have led to stronger results. Several imprecision of the previous version have been corrected to