A multi-parameter variant of the ErdH{o}s distance problem
A multi-parameter variant of the ErdH{o}s distance problem
复制标题
ErdH{o}s 距离问题的多参数变体
DOI：
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发表时间：
2017
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通讯作者：
J. Passant
中科院分区：
文献类型：
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作者：
A. Iosevich;M. Janczak;J. Passant
We study the following variant of the ErdH{o}s distance problem. Given $E$ and $F$ a point sets in $mathbb{R}^d$ and $p = (p_1, ldots, p_q)$ with $p_1+ cdots + p_q = d$ is an increasing partition of $d$ define $$ B_p(E,F)={(|x_1-y_1|, ldots, |x_q-y_q|): x in E, y in F },$$ where $x=(x_1, ldots, x_q)$ with $x_i$ in $mathbb{R}^{p_i}$. For $p_1 geq 2$ it is not difficult to construct $E$ and $F$ such that $|B_{p}(E,F)|=1$. On the other hand, it is easy to see that if $gamma_q$ is the best know exponent for the distance problem in $mathbb{R}^{p_i}$ that $|B_p(E,E)| geq C{|E|}^{frac{gamma_q}{q}}$. The question we study is whether we can improve the exponent $frac{gamma_q}{q}$. We first study partitions of length two in detail and prove the optimal result (up to logarithms) that $$ |B_{2,2}(E)| gtrapprox |E|.$$ In the generalised two dimensional case for $B_{k,l}$ we need the stronger condition that $E$ is $s$-adaptable for $s d-frac{p_1}{2}+frac{1}{3}$ we have $$ B_p(E) gtrapprox |E|^ au hspace{0.5cm} ext{where} hspace{0.5cm} au = gamma_qleft(frac{gamma_1+eta_1}{gamma_q+(q-1)(gamma_1+eta_1)}
ight).$$ Where $p_i sim frac{d}{q}$ implies $ au sim gamma_{q}left(frac{1}{q}+frac{1}{dq}
ight)$ and $p_q sim d$ (with $q<<d$) implies $ au sim gamma_{q}left(frac{1}{q}+frac{1}{q^2}
ight)$.
We study the following variant of the ErdH{o}s distance problem. Given $E$ and $F$ a point sets in $mathbb{R}^d$ and $p = (p_1, ldots, p_q)$ with $p_1+ cdots + p_q = d$ is an increasing partition of $d$ define $$ B_p(E,F)={(|x_1-y_1|, ldots, |x_q-y_q|): x in E, y in F },$$ where $x=(x_1, ldots, x_q)$ with $x_i$ in $mathbb{R}^{p_i}$. For $p_1 geq 2$ it is not difficult to construct $E$ and $F$ such that $|B_{p}(E,F)|=1$. On the other hand, it is easy to see that if $gamma_q$ is the best know exponent for the distance problem in $mathbb{R}^{p_i}$ that $|B_p(E,E)| geq C{|E|}^{frac{gamma_q}{q}}$. The question we study is whether we can improve the exponent $frac{gamma_q}{q}$. We first study partitions of length two in detail and prove the optimal result (up to logarithms) that $$ |B_{2,2}(E)| gtrapprox |E|.$$ In the generalised two dimensional case for $B_{k,l}$ we need the stronger condition that $E$ is $s$-adaptable for $s d-frac{p_1}{2}+frac{1}{3}$ we have $$ B_p(E) gtrapprox |E|^ au hspace{0.5cm} ext{where} hspace{0.5cm} au = gamma_qleft(frac{gamma_1+eta_1}{gamma_q+(q-1)(gamma_1+eta_1)}
ight).$$ Where $p_i sim frac{d}{q}$ implies $ au sim gamma_{q}left(frac{1}{q}+frac{1}{dq}
ight)$ and $p_q sim d$ (with $q<<d$) implies $ au sim gamma_{q}left(frac{1}{q}+frac{1}{q^2}
ight)$.