arXiv subjects
Publications and source records attributed to M. Z. Garaev.
For a fixed integer $n\ge 2,$ we consider the homogeneous polynomial $$ P(x_1, x_2, \ldots, x_{n+2})=\sum_{i=1}^{n} (x_2-x_1)^{i-1} x_1^{n-i} x_{i+2}. $$ We prove that, for any finite set $A$ of complex numbers, $$ \Bigl|\bigl\{P(x_1,x_2,\ldots,x_{n+2}): \, x_i\in A\bigr\}\Bigr|\gg |A|^{n}. $$ The implicit constant in $\gg$ may depend only on $n.$
Representative examples of our results are as follows. For any positive integer $N$ the equation $$ x^3+y^3=z^3+t^3, \quad x,y,z,t\in \mathbb{N}, \quad \{x,y\}\not=\{z,t\} $$ has no solutions satisfying $$ N\le x,y,z,t < N+\Bigl(\frac{38}{3}N+\frac{1297}{36}\Bigr)^{1/2}+\frac{19}{6}. $$ The strict inequality ``$<$" can not be substituted by ``$\le$", that is, there exist infinitely many positive integers $N$ such that the equation has a solution with $$ N\le x,y,z,t \le N+\Bigl(\frac{38}{3}N+\frac{1297}{36}\Bigr)^{1/2}+\frac{19}{6}. $$ There is an absolute constant $c>0$ such that for any positive integer $N$ the equation has a solution satisfying $$ N\le x,y,z,t \le N+cN^{2/3}. $$ For any $\varepsilon>0$ there exist infinitely many positive integers $N$ such that the equation has no solutions satisfying $$ N\le x,y,z,t \le N+N^{4/7-\varepsilon}. $$ There is an absolute constant $c>0$ such that for any positive integer $N$ the equation $$ x^4+y^4=z^4+t^4,\quad x,y,z,t\in\mathbb{N}, \quad \{x,y\}\not=\{z,t\}, $$ has no solutions satisfying $$ N\le x,y,z,t \le N+cN^{3/5}. $$ There is an absolute constant $c>0$ such that for any positive integer $N$ this equation has a solution satisfying $$ N\le x,y,z,t \le N+cN^{12/13}. $$
In 1997 we proved that if $n$ is of the form $$ 4k, \quad 8k-1\quad {\rm or} \quad 2^{2m+1}(2k-1)+3, $$ where $k,m\in \mathbb N,$ then there are no positive rational numbers $x,y,z$ satisfying $$ xyz = 1, \quad x+y+z = n. $$ Recently, N. X. Tho proved the following statement: let $a\in\mathbb N$ be odd and let either $n\equiv 0\pmod 4$ or $n\equiv 7\pmod 8$. Then the system of equations $$ xyz = a, \quad x+y+z = an. $$ has no solutions in positive rational numbers $x,y,z.$ A representative example of our result is the following statement: assume that $a,n\in\mathbb N$ are such that at least one of the following conditions hold: $\bullet$ $n\equiv 0\pmod 4$ $\bullet$ $n\equiv 7\pmod 8 $ $\bullet$ $a\equiv 0\pmod 4$ $\bullet$ $a\equiv 0\pmod 2$ and $n\equiv 3\pmod 4$ $\bullet$ $a^2n^3=2^{2m+1}(2k-1)+27$ for some $k,m\in \mathbb N.$ Then the system of equations $$ xyz = a, \quad x+y+z = an. $$ has no solutions in positive rational numbers $x,y,z.$
Let $p$ be a large prime number and $g$ be any integer of multiplicative order $T$ modulo $p$. We obtain a new estimate of the double exponential sum $$ S=\sum_{n\in \mathcal{N}}\left|\sum_{m\in \mathcal{M} }e_p(an g^{m})\right|, \quad \gcd (a,p)=1, $$ where $\mathcal{N}$ and $\mathcal{M}$ are intervals of consecutive integers with $|\mathcal{N}|=N$ and $|\mathcal{M}|=M<T$ elements. One representative example is the following consequence of the main result: if $N=M\approx p^{1/3}$, then $|S|< N^{2-1/8 + o(1)}$. We then apply our estimate to obtain new results on additive congruences involving intervals and exponential functions.
We prove several results which imply the following consequences. For any $\varepsilon>0$ and any sufficiently large prime $p$, if $\cI_1,\ldots, \cI_{13}$ are intervals of cardinalities $|\cI_j|>p^{1/4+\varepsilon}$ and $abc\not\equiv 0\pmod p$, then the congruence $$ ax_1\cdots x_6+bx_7\cdots x_{13}\equiv c\pmod p $$ has a solution with $x_j\in\cI_j$. There exists an absolute constant $n_0\in\N$ such that for any $0<\varepsilon<1$ and any sufficiently large prime $p$, any quadratic residue $λ$ modulo $p$ can be represented in the form $$ x_1\cdots x_{n_0}\equiv λ\pmod p,\quad x_i\in\N,\quad x_i\le p^{1/(4e^{2/3})+\varepsilon}. $$ For any $\varepsilon>0$ there exists $n=n(\varepsilon)\in \N$ such that for any sufficiently large $m\in\N$ the congruence $$ x_1\cdots x_{n}\equiv 1\pmod m,\quad x_i\in\N,\quad x_i\le m^{\varepsilon} $$ has a solution with $x_1\not=1$.
Let $\F_p$ be the field of residue classes modulo a large prime $p$. The present paper is devoted to the problem of representability of elements of $\F_p$ as sums of fractions of the form $x/y$ with $x,y$ from short intervals of $\F_p$.
Let $p$ be a prime, $\varepsilon>0$ and $0 c (N\log N)^{1/2},\,\, c=c(\varepsilon)>0. $$ We use this bound to show that any $λ\not\equiv 0\pmod p$ can be represented in the form $λ\equiv n_1!...n_7!\pmod p$, where $n_i=o(p^{11/12})$. This slightly refines the previously known range for $n_i$.
In the present paper we obtain new upper bound estimates for the number of solutions of the congruence $$ x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in\cU, $$ for certain ranges of $H$ and $|\cU|$, where $\cU$ is a subset of the field of residue classes modulo $p$ having small multiplicative doubling. We then use this estimate to show that the number of solutions of the congruence $$ x^n\equiv λ\pmod p; \quad x\in \N, \quad L 0$. This implies, in particular, that if $f(x)\in \Z[x]$ is a fixed polynomial without multiple roots in $\C$, then the congruence $ x^{f(x)}\equiv 1\pmod p, \,x\in \mathbb{N}, \,x\le p,$ has at most $p^{\frac{1}{3}-c}$ solutions as $p\to\infty$, improving some recent results of Kurlberg, Luca and Shparlinski and of Balog, Broughan and Shparlinski. We use our results to show that almost all the residue classes modulo $p$ can be represented in the form $xg^y \pmod p$ with positive integers $x<p^{5/8+\varepsilon}$ and $y<p^{3/8}$. Here $g$ denotes a primitive root modulo $p$. We also prove that almost all the residue classes modulo $p$ can be represented in the form $xyzg^t \pmod p$ with positive integers $x,y,z,t<p^{1/4+\varepsilon}$.
We prove that there exists an absolute constant $c>0$ such that if an arithmetic progression $\cP$ modulo a prime number $p$ does not contain zero and has the cardinality less than $cp$, then it can not be represented as a product of two subsets of cardinality greater than 1, unless $\cP=-\cP$ or $\cP=\{-2r,r,4r\}$ for some residue $r$ modulo $p$.
In the present paper, we generalize some of the results on Kloosterman sums proven in \cite{BG} for prime moduli to general moduli. This requires to establish the corresponding additive properties of the reciprocal set $$ I^{-1}=\{x^{-1}:\quad x\in I\}, $$ where $I$ is an interval in the ring of residue classes modulo a large positive integer. We apply our bounds on multilinear exponential sums to the Brun-Titchmarsh theorem and the estimate of very short Kloosterman sums, hence generalizing our earlier work to the setting of general modulus.
We obtain new results on additive properties of the set $$ I^{-1}= \{x^{-1}: \quad x\in I\} $$ where $I$ is an arbitrary interval in the field of residue classes modulo a large prime $p$. We combine our results with multilinear exponential sum estimates and obtain new results on incomplete multilinear Kloosterman sums.
Let $p$ be a large prime number, $K,L,M,λ$ be integers with $1\le M\le p$ and ${\color{red}\gcd}(λ,p)=1.$ The aim of our paper is to obtain sharp upper bound estimates for the number $I_2(M; K,L)$ of solutions of the congruence $$ xy\equivλ\pmod p, \qquad K+1\le x\le K+M,\quad L+1\le y\le L+M $$ and for the number $I_3(M;L)$ of solutions of the congruence $$xyz\equivλ\pmod p, \quad L+1\le x,y,z\le L+M. $$ We obtain a bound for $I_2(M;K,L),$ which improves several recent results of Chan and Shparlinski. For instance, we prove that if $M<p^{1/4},$ then $I_2(M;K,L)\le M^{o(1)}.$ For $I_3(M;L)$ we prove that if $M<p^{1/8}$ then $I_3(M;L)\le M^{o(1)}.$ Our results have applications to some other problems as well. For instance, it follows that if $\mathcal{I}_1, \mathcal{I}_2, \mathcal{I}_3$ are intervals in $\F^*_p$ of length $|\mathcal{I}_i|< p^{1/8},$ then $$ |\mathcal{I}_1\cdot \mathcal{I}_2\cdot \mathcal{I}_3|= (|\mathcal{I}_1|\cdot |\mathcal{I}_2|\cdot |\mathcal{I}_3|)^{1-o(1)}. $$
Let $F_p$ be the field of a prime order $p.$ For a subset $A\subset F_p$ we consider the product set $A(A+1).$ This set is an image of $A\times A$ under the polynomial mapping $f(x,y)=xy+x:F_p\times F_p\to F_p.$ In the present paper we show that if $|A| p^{2/3},$ then we prove that $$|A(A+1)|\gg \sqrt{p |A|}$$ and show that this is the optimal in general settings bound up to the implied constant. We also estimate the cardinality of $A(A+1)$ when $A$ is a subset of real numbers. We show that in this case one has the Elekes type bound $$ |A(A+1)|\gg |A|^{5/4}. $$
Let $ε$ be a fixed positive quantity, $m$ be a large integer, $x_j$ denote integer variables. We prove that for any positive integers $N_1,N_2,N_3$ with $N_1N_2N_3>m^{1+ε},$ the set $$ \{x_1x_2x_3 \pmod m: \quad x_j\in [1,N_j] \} $$ contains almost all the residue classes modulo $m$ (i.e., its cardinality is equal to $m+o(m)$). We further show that if $m$ is cubefree, then for any positive integers $N_1,N_2,N_3,N_4$ with $N_1N_2N_3N_4>m^{1+ε},$ the set $$ \{x_1x_2x_3x_4 \pmod m: \quad x_j\in [1,N_j] \} $$ also contains almost all the residue classes modulo $m.$ Let $p$ be a large prime parameter and let $p>N>p^{63/76+ε}.$ We prove that for any nonzero integer constant $k$ and any integer $λ\not\equiv 0\pmod p$ the congruence $$ p_1p_2(p_3+k)\equiv λ\pmod p $$ admits $(1+o(1))π(N)^3/p$ solutions in prime numbers $p_1, p_2, p_3\le N.$
Let $q$ be a large prime number, $a$ be any integer, $ε$ be a fixed small positive quantity. Friedlander and Shparlinksi \cite{FSh} have shown that there exists a positive integer $n\ll q^{5/2+ε}$ such that $ϕ(n)$ falls into the residue class $a \pmod q.$ Here, $ϕ(n)$ denotes Euler's function. In the present paper we improve this bound to $n\ll q^{2+ε}.$
We show that for any fixed $\eps>0$, there are numbers $δ>0$ and $p_0\ge 2$ with the following property: for every prime $p\ge p_0$ and every integer $N$ such that $p^{1/(4\sqrt{e})+\eps}\le N\le p$, the sequence $1,2,...,N$ contains at least $δN$ quadratic non-residues modulo $p$. We use this result to obtain strong upper bounds on the sizes of the least quadratic non-residues in Beatty and Piatetski--Shapiro sequences.
Let $\mathbb{F}_p$ be the field of a prime order $p.$ It is known that for any integer $N\in [1,p]$ one can construct a subset $A\subset\mathbb{F}_p$ with $|A|= N$ such that $$ \max\{|A+A|, |AA|\}\ll p^{1/2}|A|^{1/2}. $$ In the present paper we prove that if $A\subset \mathbb{F}_p$ with $|A|>p^{2/3},$ then $$ \max\{|A+A|, |AA|\}\gg p^{1/2}|A|^{1/2}. $$
We prove that for any zero $β'+iγ'$ of $ζ'(s)$ there exists a zero $β+iγ$ of $ζ(s)$ such that $|γ-γ'|\ll \sqrt{|β'-\tfrac{1}{2}|},$ and we provide some other related results.