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Liangpan Li

Publications and source records attributed to Liangpan Li.

24 records · Page 2Linked to original sources
arXiv2009

Slightly improved sum-product estimates in fields of prime order

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.

math.NT↗
arXiv2008

Collinear triples in permutations

Let $α:\mathbb{F}_q\to\mathbb{F}_q$ be a permutation and $Ψ(α)$ be the number of collinear triples in the graph of $α$, where $\mathbb{F}_q$ denotes a finite field of $q$ elements. When $q$ is odd Cooper and Solymosi once proved $Ψ(α)\geq(q-1)/4$ and conjectured the sharp bound should be $Ψ(α)\geq(q-1)/2$. In this note we indicate that the Cooper-Solymosi conjecture is true.

math.CO↗
arXiv2008

On the number of collinear triples in permutations

Let $α:\mathbb{Z}_n\to\mathbb{Z}_n$ be a permutation and $Ψ(α)$ be the number of collinear triples modulo $n$ in the graph of $α$. Cooper and Solymosi had given by induction the bound $\min_αΨ(α)\geq\lceil(n-1)/4\rceil$ when $n$ is a prime number. The main purpose of this paper is to give a direct proof of that bound. Besides, the expected number of collinear triples a permutation can have is also been determined.

math.CO↗
arXiv2008

On the size of Nikodym sets in finite fields

Let $\mathbb{F}_q$ denote a finite field of $q$ elements. Define a set $B\subset\mathbb{F}_q^n$ to be Nikodym if for each $x\in B^{c}$, there exists a line $L$ such that $L\cap B^c=\{x\}.$ The main purpose of this note is to show that the size of every Nikodym set is at least $C_n\cdot q^n$, where $C_n$ depends only on $n$.

math.CA↗
arXiv2007

The Graham conjecture implies the Erdos-Turan conjecture

Erdös and Turán once conjectured that any set $A\subset\mathbb{N}$ with $\sum_{a\in A}{1}/{a}=\infty$ should contain infinitely many progressions of arbitrary length $k\geq3$. For the two-dimensional case Graham conjectured that if $B\subset \mathbb{N}\times\mathbb{N}$ satisfies $$\sum\limits_{(x,y)\in B}\frac{1}{x^2+y^2}=\infty,$$ then for any $s\geq2$, $B$ contains an $s\times s$ axes-parallel grid. In this paper it is shown that if the Graham conjecture is true for some $s\geq2$, then the Erdös-Turán conjecture is true for $k=2s-1$.

math.NT↗