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Comment: 4 pages
Let $\mathbb{F}_p$ be the field of residue classes modulo a prime number $p$ and let $A$ be a nonempty subset of $\mathbb{F}_p$. In this paper we show that if $|A|\preceq p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succeq|A|^{13/12};\] if $|A|\succeq p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succapprox \min\{|A|^{13/12}(\frac{|A|}{p^{0.5}})^{1/12},|A|(\frac{p}{|A|})^{1/11}\}.\] These results slightly improve the estimates of Bourgain-Garaev and Shen. Sum-product estimates on different sets are also considered.
In this paper we provide a complete approach to the real numbers via decimal representations. Construction of the real numbers by Dedekind cuts, Cauchy sequences of rational numbers, and the algebraic characterization of the real number system by the concept of complete ordered field are also well explained in the new setting.
In this paper we show that for any $k\geq2$, there exist two universal constants $C_k,D_k>0$, such that for any finite subset $A$ of positive real numbers with $|AA|\leq M|A|$, $|kA|\geq \frac{C_k}{M^{D_k}}\cdot|A|^{\log_42k}.$
A set of reals $A=\{a_1,...,a_n\}$ labeled in increasing order is called convex if there exists a continuous strictly convex function $f$ such that $f(i)=a_i$ for every $i$. Given a convex set $A$, we prove \[|A+A|\gg\frac{|A|^{14/9}}{(\log|A|)^{2/9}}.\] Sumsets of different summands and an application to a sum-product-type problem are also studied either as remarks or as theorems.
Answering two questions of Beresnevich and Velani, we develop zero-one laws in both simultaneous and multiplicative Diophantine approximation. Our proofs rely on a Cassels-Gallagher type theorem as well as a higher-dimensional analogue of the cross fibering principle of Beresnevich, Haynes and Velani.
Let $A$ be a finite set of positive real numbers. We present a sum-division estimate: |A+A|^2|A/A|\geq\frac{|A|^4}{4}.
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