Ramsey-type theorems for sets satisfying a geometric regularity condition

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• 作者：P. Hagelstein ; ¹ ; ^{paul_hagelstein@baylor.edu" class="auth_mail" title="E-mail the corresponding author} ; D. Herden ^{daniel_herden@baylor.edu" class="auth_mail" title="E-mail the corresponding author} ; D. Young² ; ^{daniel_young1@baylor.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Ramsey theory ; Regularity condition ; Maximal operators
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：447
• 期：2
• 页码：951-956
• 全文大小：291 K

文摘

We consider Ramsey-type problems associated to collections of sets in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306436&_mathId=si1.gif&_user=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=2d954dd1e588cb5ae7e7640c55e897ae" title="Click to view the MathML source">Rⁿclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^n$ satisfying a standard geometric regularity condition. In particular, let class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306436&_mathId=si2.gif&_user=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=46ee81650a7453b0d925f4c039ba4f86">class="imgLazyJSB inlineImage" height="21" width="58" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306436-si2.gif">class="mathContainer hidden">class="mathCode">${\{R_j\}}_{j=1}^N$ be a collection of measurable sets in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306436&_mathId=si1.gif&_user=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=2d954dd1e588cb5ae7e7640c55e897ae" title="Click to view the MathML source">Rⁿclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^n$ such that every class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306436&_mathId=si3.gif&_user=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=c56f6f6095ac169966f7a004cb7d3e20" title="Click to view the MathML source">R_jclass="mathContainer hidden">class="mathCode">$R_j$ is contained in a cube class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306436&_mathId=si4.gif&_user=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=2c9619c098ef77fbe193ed88030ec2c2" title="Click to view the MathML source">Q_jclass="mathContainer hidden">class="mathCode">$Q_j$ whose sides are parallel to the axes and such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306436&_mathId=si5.gif&_user=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=b1752a799947940fdc01160c90812b10" title="Click to view the MathML source">|R_j|/|Q_j|≥ρ>0class="mathContainer hidden">class="mathCode">$|R_j|/|Q_j|\ge \rho >0$. Moreover, suppose that there exists class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306436&_mathId=si6.gif&_user=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=95712db7778f6a09483cae1ce281a773" title="Click to view the MathML source">0<γ<∞class="mathContainer hidden">class="mathCode">$0<\gamma <\infty$ such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306436&_mathId=si7.gif&_user=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=93ee0e433a34221d292360089d2d6201" title="Click to view the MathML source">|R_j|/|R_k|≤γclass="mathContainer hidden">class="mathCode">$|R_j|/|R_k|\le \gamma$ for every class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306436&_mathId=si8.gif&_user=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=ebb206b67a70e850759c100c069abfee" title="Click to view the MathML source">j,kclass="mathContainer hidden">class="mathCode">$j,k$. We prove that there exists a subcollection of class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306436&_mathId=si2.gif&_user=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=46ee81650a7453b0d925f4c039ba4f86">class="imgLazyJSB inlineImage" height="21" width="58" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306436-si2.gif">class="mathContainer hidden">class="mathCode">${\{R_j\}}_{j=1}^N$ consisting of at least class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306436&_mathId=si37.gif&_user=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=a0dd968377980c227587eda730175783" title="Click to view the MathML source">R(N)class="mathContainer hidden">class="mathCode">$R(N)$ sets that either have a point of common intersection or that are pairwise disjoint, where class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306436&_mathId=si10.gif&_user=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=1150b1c97d4d9893e8fad4e53e382d1f">class="imgLazyJSB inlineImage" height="34" width="177" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306436-si10.gif">class="mathContainer hidden">class="mathCode">$R(N)\ge {\left(\frac{N\rho }{{(1+2\cdot {\gamma }^{1/n})}^n}\right)}^{1/2}$. If the sets in the collection class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306436&_mathId=si11.gif&_user=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=34a501a5e5a180f05ba11025412b4e9a" title="Click to view the MathML source">{R_j}class="mathContainer hidden">class="mathCode">$\{R_j\}$ are convex, we obtain the improved Ramsey estimate class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306436&_mathId=si12.gif&_user=111111111&_pii=S0022247X16306436&_rdoc=1&_issn=0022247X&md5=66f9d48073a0c53f5abf275cbd3e6a69" title="Click to view the MathML source">R(N)≥(3⁻ⁿρN)^1/2class="mathContainer hidden">class="mathCode">$R(N)\ge {\left(3^{-n}\rho N\right)}^{1/2}$. Applications of these results to weak type bounds of geometric maximal operators are provided.