Related papers: Sum-product estimates for rational functions
We improve a previous sum--products estimates in R, namely, we obtain that max{|A+A|,|AA|} \gg |A|^{4/3+c}, where c any number less than 5/9813. New lower bounds for sums of sets with small the product set are found. Also we prove some pure…
Let $K$ be a number field. This paper considers arithmetic functions over $K$, that are, complex valued functions on the set of nonzero integral ideals in $K$. Firstly we generalize some basic results on arithmetic functions. Next we define…
In this paper we further study the relationship between convexity and additive growth, building on the work of Schoen and Shkredov (\cite{SS}) to get some improvements to earlier results of Elekes, Nathanson and Ruzsa (\cite{ENR}). In…
We construct families of explicit polynomials f with rational coefficients that are sums of squares of polynomials over the real numbers, but not over the rational numbers. Whether or not such examples exist was an open question originally…
The authors review results implicit in their recent paper [2] on the product/quotient representation of rationals by rationals of the type $( an + b )/ ( An+ B )$ and give a detailed account of a particular related non-intuitive…
This paper considers various formulations of the sum-product problem. It is shown that, for a finite set $A\subset{\mathbb{R}}$, $$|A(A+A)|\gg{|A|^{\frac{3}{2}+\frac{1}{178}}},$$ giving a partial answer to a conjecture of Balog. In a…
A rational function $f(x)$ is rationally summable if there exists a rational function $g(x)$ such that $f(x)=g(x+1)-g(x)$. Detecting whether a given rational function is summable is an important and basic computational subproblem that…
We find all polynomials f,g,h over a field K such that g and h are linear and f(g(x))=h(f(x)). We also solve the same problem for rational functions f,g,h, in case the field K is algebraically closed.
The paper considers estimates for some sums and products of functions of prime numbers. Several assertions on this topic have been proven. We also study extremal estimates for strongly additive and strongly multiplicative arithmetic…
Let $\mathcal R$ be a finite valuation ring of order $q^r$ with $q$ a power of an odd prime number, and $\mathcal A$ be a set in $\mathcal R$. In this paper, we improve a recent result due to Yazici (2018) on a sum-product type problem.…
Given a suitable arithmetic function h, we investigate the average order of h as it ranges over the values taken by an integral binary form F. A general upper bound is obtained for this quantity, in which the dependence upon the…
We investigate fractional sums of arithmetic functions over products of two or three integers, with emphasis on fixed greatest common divisors and multiplicative weights. Let $f$ be an arithmetic function satisfying $f(n) \ll n^\alpha$ for…
In this paper we provide in $\bFp$ expanding lower bounds for two variables functions $f(x,y)$ in connection with the product set or the sumset. The sum-product problem has been hugely studied in the recent past. A typical result in…
We study the $\delta$-discretized sum-product estimates for well spaced sets. Our main result is: for a fixed $\alpha\in(1,\frac{3}{2}]$, we prove that for any $\sim|A|^{-1}$-separated set $A\subset[1,2]$ and $\delta=|A|^{-\alpha}$, we…
Let A be a class of objects, equipped with an integer size such that for all n the number a(n) of objects of size n is finite. We are interested in the case where the generating fucntion sum_n a(n) t^n is rational, or more generally…
A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of the Fa\`a di Bruno's formula.…
Let $\mathbb{F}_q$ denote the finite field with $q$ elements where $q=p^l$ is a prime power. Using Fourier analytic tools with a third moment method, we obtain sum-product type estimates for subsets of $\mathbb{F}_q$. In particular, we…
In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical…
If g and h are functions over some field, we can consider their composition f = g(h). The inverse problem is decomposition: given f, determine the ex- istence of such functions g and h. In this thesis we consider functional decom- positions…
For various arithmetic functions $f:\mathbb{N} \to \mathbb{R}$, the behavior of $f(n!)$ and that of $\sum_{n\le N} f(n!)$ can be intriguing. For instance, for some functions $f$, we have ${f(n!)=\sum_{k\le n}f(k)}$, for others, we have…