Related papers: Sum-product estimates for rational functions

Polynomial functions $f : \mathbb{N}_+ \longrightarrow \mathbb{N}_+$ are studied for which sums of arbitrary length $f (1) + f (2) + f (3) + >... + f (n)$, with $n \in \mathbb{N}_+$, can be expressed by polynomial functions $g :…

Let $\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\F_q[x_1, x_2]$. For a set $A \subset \F_q$, define $P(A):=\{P(x_1, x_2) | x_i \in A \}$. Using certain constructions of expanders, we characterize all polynomials $P$…

Let $\Delta_x f(x,y)=f(x+1,y)-f(x,y)$ and $\Delta_y f(x,y)=f(x,y+1)-f(x,y)$ be the difference operators with respect to $x$ and $y$. A rational function $f(x,y)$ is called summable if there exist rational functions $g(x,y)$ and $h(x,y)$…

The aim of this note is to record a proof that the estimate $$\max{\{|A+A|,|A:A|\}}\gg{|A|^{12/11}}$$ holds for any set $A\subset{\mathbb{F}_q}$, provided that $A$ satisfies certain conditions which state that it is not too close to being a…

A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite…

We classify the polynomials $f(x,y) \in \mathbb R[x,y]$ such that given any finite set $A \subset \mathbb R$ if $|A+A|$ is small, then $|f(A,A)|$ is large. In particular, the following bound holds : $|A+A||f(A,A)| \gtrsim |A|^{5/2}.$ The…

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…

Let $A$ be a finite subset of $\ffield$, the field of Laurent series in $1/t$ over a finite field $\mathbb{F}_q$. We show that for any $\epsilon>0$ there exists a constant $C$ dependent only on $\epsilon$ and $q$ such that…

Let $f$ be a smooth real function with strictly monotone first $k$ derivatives. We show that for a finite set $A$, with $|A+A|\leq K|A|$, $|2^kf(A)-(2^k-1)f(A)|\gg_k |A|^{k+1-o(1)}/K^{O_k(1)}$. We deduce several new sum-product type…

We present criteria for deciding whether a bivariate rational function in two variables can be written as a sum of two (q-)differences of bivariate rational functions. Using these criteria, we show how certain double sums can be evaluated,…

We estimate the sum of products or quotients of $L$-functions, where the sum is taken over all quadratic extensions of given genus over a fixed global function field. Our estimate for the sum of the quotient of two $L$-functions is…

In this paper we prove some results on sum-product estimates over arbitrary finite fields. More precisely, we show that for sufficiently small sets $A\subset \mathbb{F}_q$ we have \[|(A-A)^2+(A-A)^2|\gg |A|^{1+\frac{1}{21}}.\] This can be…

Let $\mathbb{A}=\mathbb{F}_{q}[T]$ be the polynomial ring over finite field $\mathbb{F}_{q}$, and $\mathbb{A}_{+}$ be the set of monic polynomials in $\mathbb{A}$. In this paper, we show that a large class of arithmetic functions in…

In this paper we prove new bounds for sums of convex or concave functions. Specifically, we prove that for all $A,B \subseteq \mathbb R$ finite sets, and for all $f,g$ convex or concave functions, we have $$|A + B|^{38}|f(A) + g(B)|^{38}…

The paper proves sum-of-square-of-rational-function based representations (shortly, sosrf-based representations) of polynomial matrices that are positive semidefinite on some special sets: $\mathbb{R}^n;$ $\mathbb{R}$ and its intervals…

Let $\mathbb{F}_p$ be the field of residue classes modulo a prime number $p$ and let $A$ be a non-empty subset of $\mathbb{F}_p.$ In this paper we give an explicit version of the sum-product estimate of Bourgain, Katz, Tao and Bourgain,…

Expanding upon recent work, a new class of $A$-functions is introduced that can be viewed as an appropriate generalization of the class of regular $A$-functions, the class of structured $A$-functions, and the class of perfect $A$-functions.…

The \emph{sum-product phenomenon} predicts that a finite set $A$ in a ring $R$ should have either a large sumset $A+A$ or large product set $A \cdot A$ unless it is in some sense "close" to a finite subring of $R$. This phenomenon has been…

Let q be a prime, A be a subset of a finite field $F=\Bbb Z/q\Bbb Z$, $|A|<\sqrt{|F|}$. We prove the estimate $\max(|A+A|,|A\cdot A|)\ge c|A|^{1+\epsilon}$ for some $\epsilon>0$ and c>0. This extends the result of J. Bourgain, N. Katz, and…

Given $d \in \mathbb{N}$, we establish sum-product estimates for finite, non-empty subsets of $\mathbb{R}^d$. This is equivalent to a sum-product result for sets of diagonal matrices. In particular, let $A$ be a finite, non-empty set of $d…