Pub Date : 2021-08-12DOI: 10.1017/s0305004121000578
{"title":"PSP volume 171 issue 2 Cover and Back matter","authors":"","doi":"10.1017/s0305004121000578","DOIUrl":"https://doi.org/10.1017/s0305004121000578","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73606596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-12DOI: 10.1017/s0305004121000566
{"title":"PSP volume 171 issue 2 Cover and Front matter","authors":"","doi":"10.1017/s0305004121000566","DOIUrl":"https://doi.org/10.1017/s0305004121000566","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82906387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-03DOI: 10.1017/S0305004122000317
Natalia S. Dergacheva, A. Klyachko
Abstract How many 2-cells must two finite CW-complexes have to admit a common, but not finite common, covering? Leighton’s theorem says that both complexes must have 2-cells. We construct an almost (?) minimal example with two 2-cells in each complex.
{"title":"Small non-Leighton two-complexes","authors":"Natalia S. Dergacheva, A. Klyachko","doi":"10.1017/S0305004122000317","DOIUrl":"https://doi.org/10.1017/S0305004122000317","url":null,"abstract":"Abstract How many 2-cells must two finite CW-complexes have to admit a common, but not finite common, covering? Leighton’s theorem says that both complexes must have 2-cells. We construct an almost (?) minimal example with two 2-cells in each complex.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72552053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-26DOI: 10.1017/S0305004123000166
T. Wooley
Abstract When $kgeqslant 4$ and $0leqslant dleqslant (k-2)/4$ , we consider the system of Diophantine equations begin{align*}x_1^j+ldots +x_k^j=y_1^j+ldots +y_k^jquad (1leqslant jleqslant k,, jne k-d).end{align*} We show that in this cousin of a Vinogradov system, there is a paucity of non-diagonal positive integral solutions. Our quantitative estimates are particularly sharp when $d=o!left(k^{1/4}right)$ .
{"title":"Paucity problems and some relatives of Vinogradov’s mean value theorem","authors":"T. Wooley","doi":"10.1017/S0305004123000166","DOIUrl":"https://doi.org/10.1017/S0305004123000166","url":null,"abstract":"Abstract When \u0000$kgeqslant 4$\u0000 and \u0000$0leqslant dleqslant (k-2)/4$\u0000 , we consider the system of Diophantine equations \u0000begin{align*}x_1^j+ldots +x_k^j=y_1^j+ldots +y_k^jquad (1leqslant jleqslant k,, jne k-d).end{align*}\u0000 We show that in this cousin of a Vinogradov system, there is a paucity of non-diagonal positive integral solutions. Our quantitative estimates are particularly sharp when \u0000$d=o!left(k^{1/4}right)$\u0000 .","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77273162","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-22DOI: 10.1017/S0305004122000123
Claudemir Fidelis, P. Koshlukov
Abstract Let K be any field of characteristic two and let $U_1$ and $W_1$ be the Lie algebras of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$ and of the polynomial ring K[t], respectively. The algebras $U_1$ and $W_1$ are equipped with natural $mathbb{Z}$ -gradings. In this paper, we provide bases for the graded identities of $U_1$ and $W_1$ , and we prove that they do not admit any finite basis.
{"title":"$mathbb{Z}$\u0000 -graded identities of the Lie algebras \u0000$U_1$\u0000 in characteristic 2","authors":"Claudemir Fidelis, P. Koshlukov","doi":"10.1017/S0305004122000123","DOIUrl":"https://doi.org/10.1017/S0305004122000123","url":null,"abstract":"Abstract Let K be any field of characteristic two and let \u0000$U_1$\u0000 and \u0000$W_1$\u0000 be the Lie algebras of the derivations of the algebra of Laurent polynomials \u0000$K[t,t^{-1}]$\u0000 and of the polynomial ring K[t], respectively. The algebras \u0000$U_1$\u0000 and \u0000$W_1$\u0000 are equipped with natural \u0000$mathbb{Z}$\u0000 -gradings. In this paper, we provide bases for the graded identities of \u0000$U_1$\u0000 and \u0000$W_1$\u0000 , and we prove that they do not admit any finite basis.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81806569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-15DOI: 10.1017/S0305004121000682
A. Berger, A. Sah, Mehtaab Sawhney, Jonathan Tidor
Abstract In this paper we characterize when non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$ -norm. We give a brief deduction of the fact that a bounded function on $mathbb F_p^n$ with large $U^k$ -norm must correlate with a classical polynomial when $kle p+1$ . To the best of our knowledge, this result is new for $k=p+1$ (when $p>2$ ). We then prove that non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$ -norm over $mathbb F_p^n$ for all $kge p+2$ , completely characterising when classical polynomials suffice.
{"title":"Non-classical polynomials and the inverse theorem","authors":"A. Berger, A. Sah, Mehtaab Sawhney, Jonathan Tidor","doi":"10.1017/S0305004121000682","DOIUrl":"https://doi.org/10.1017/S0305004121000682","url":null,"abstract":"Abstract In this paper we characterize when non-classical polynomials are necessary in the inverse theorem for the Gowers \u0000$U^k$\u0000 -norm. We give a brief deduction of the fact that a bounded function on \u0000$mathbb F_p^n$\u0000 with large \u0000$U^k$\u0000 -norm must correlate with a classical polynomial when \u0000$kle p+1$\u0000 . To the best of our knowledge, this result is new for \u0000$k=p+1$\u0000 (when \u0000$p>2$\u0000 ). We then prove that non-classical polynomials are necessary in the inverse theorem for the Gowers \u0000$U^k$\u0000 -norm over \u0000$mathbb F_p^n$\u0000 for all \u0000$kge p+2$\u0000 , completely characterising when classical polynomials suffice.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83875875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-07DOI: 10.1017/S0305004121000360
Yiftach Dayan
We show that fractal percolation sets in $mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $Esubsetmathbb{R}^{d}$ is a realisation of a fractal percolation process, then almost surely (conditioned on $Eneqemptyset$ ), for every countable collection $left(f_{i}right)_{iinmathbb{N}}$ of $C^{1}$ diffeomorphisms of $mathbb{R}^{d}$ , $dim_{H}left(Ecapleft(bigcap_{iinmathbb{N}}f_{i}left(text{BA}_{d}right)right)right)=dim_{H}left(Eright)$ , where $text{BA}_{d}$ is the set of badly approximable vectors in $mathbb{R}^{d}$ . We show this by proving that E almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to $dim_{H}left(Eright)$ . We achieve this by analysing Galton–Watson trees and showing that they almost surely contain appropriate subtrees whose projections to $mathbb{R}^{d}$ yield the aforementioned subsets of E. This method allows us to obtain a more general result by projecting the Galton–Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.
{"title":"Random fractals and their intersection with winning sets","authors":"Yiftach Dayan","doi":"10.1017/S0305004121000360","DOIUrl":"https://doi.org/10.1017/S0305004121000360","url":null,"abstract":"\u0000 We show that fractal percolation sets in \u0000 \u0000 \u0000 $mathbb{R}^{d}$\u0000 \u0000 almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if \u0000 \u0000 \u0000 $Esubsetmathbb{R}^{d}$\u0000 \u0000 is a realisation of a fractal percolation process, then almost surely (conditioned on \u0000 \u0000 \u0000 $Eneqemptyset$\u0000 \u0000 ), for every countable collection \u0000 \u0000 \u0000 $left(f_{i}right)_{iinmathbb{N}}$\u0000 \u0000 of \u0000 \u0000 \u0000 $C^{1}$\u0000 \u0000 diffeomorphisms of \u0000 \u0000 \u0000 $mathbb{R}^{d}$\u0000 \u0000 , \u0000 \u0000 \u0000 $dim_{H}left(Ecapleft(bigcap_{iinmathbb{N}}f_{i}left(text{BA}_{d}right)right)right)=dim_{H}left(Eright)$\u0000 \u0000 , where \u0000 \u0000 \u0000 $text{BA}_{d}$\u0000 \u0000 is the set of badly approximable vectors in \u0000 \u0000 \u0000 $mathbb{R}^{d}$\u0000 \u0000 . We show this by proving that E almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to \u0000 \u0000 \u0000 $dim_{H}left(Eright)$\u0000 \u0000 .\u0000 We achieve this by analysing Galton–Watson trees and showing that they almost surely contain appropriate subtrees whose projections to \u0000 \u0000 \u0000 $mathbb{R}^{d}$\u0000 \u0000 yield the aforementioned subsets of E. This method allows us to obtain a more general result by projecting the Galton–Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86649696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-01DOI: 10.1017/s030500412100044x
{"title":"PSP volume 171 issue 1 Cover and Front matter","authors":"","doi":"10.1017/s030500412100044x","DOIUrl":"https://doi.org/10.1017/s030500412100044x","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84513931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-01DOI: 10.1017/s0305004121000451
{"title":"PSP volume 171 issue 1 Cover and Back matter","authors":"","doi":"10.1017/s0305004121000451","DOIUrl":"https://doi.org/10.1017/s0305004121000451","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85629124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-16DOI: 10.1017/S0305004122000147
M. Buci'c, Oliver Janzer, B. Sudakov
Abstract In 1974, Erdős posed the following problem. Given an oriented graph H, determine or estimate the maximum possible number of H-free orientations of an n-vertex graph. When H is a tournament, the answer was determined precisely for sufficiently large n by Alon and Yuster. In general, when the underlying undirected graph of H contains a cycle, one can obtain accurate bounds by combining an observation of Kozma and Moran with celebrated results on the number of F-free graphs. As the main contribution of the paper, we resolve all remaining cases in an asymptotic sense, thereby giving a rather complete answer to Erdős’s question. Moreover, we determine the answer exactly when H is an odd cycle and n is sufficiently large, answering a question of Araújo, Botler and Mota.
{"title":"Counting H-free orientations of graphs","authors":"M. Buci'c, Oliver Janzer, B. Sudakov","doi":"10.1017/S0305004122000147","DOIUrl":"https://doi.org/10.1017/S0305004122000147","url":null,"abstract":"Abstract In 1974, Erdős posed the following problem. Given an oriented graph H, determine or estimate the maximum possible number of H-free orientations of an n-vertex graph. When H is a tournament, the answer was determined precisely for sufficiently large n by Alon and Yuster. In general, when the underlying undirected graph of H contains a cycle, one can obtain accurate bounds by combining an observation of Kozma and Moran with celebrated results on the number of F-free graphs. As the main contribution of the paper, we resolve all remaining cases in an asymptotic sense, thereby giving a rather complete answer to Erdős’s question. Moreover, we determine the answer exactly when H is an odd cycle and n is sufficiently large, answering a question of Araújo, Botler and Mota.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89296972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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