\documentclass[CRMATH,Unicode,biblatex,XML]{cedram} \TopicFR{Systèmes dynamiques} \TopicEN{Dynamical systems} \TopicFR{Théorie des nombres} \TopicEN{Number theory} \addbibresource{CRMATH_Hussain_20230199.bib} \usepackage{amssymb} \newenvironment{problem}{\begin{enonce}{Problem}}{\end{enonce}} \newenvironment{Obstruction}{\begin{enonce}{Obstruction}}{\end{enonce}} \newcommand{\DD}{\mathcal D} \newcommand{\HH}{\mathcal H} \newcommand{\MM}{\mathcal M} \newcommand{\bfP}{\mathbf P} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R} \newcommand{\Z}{\mathbb Z} \newcommand{\mbf}{\mathbf} \newcommand{\0}{\mbf 0}% Moved since \00 is not a valid macro \newcommand{\ee}{\mbf e} \newcommand{\qq}{\mbf q} \newcommand{\rr}{\mbf r} \newcommand{\vv}{\mbf v} \newcommand{\xx}{\mbf x} \newcommand{\yy}{\mbf y} \newcommand{\zz}{\mbf z} \newcommand{\WWW}{W_{n}^{\theta}(\Psi)} \newcommand{\DDD}{\DD_n^{\theta}(\Psi)} \newenvironment{GBSP1}{\begin{enonce*}{Generalised Baker--Schmidt Problem for Hausdorff Measure: dual setting}}{\end{enonce*}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \makeatletter \let\save@mathaccent\mathaccent \newcommand*\if@single[3]{% \setbox0\hbox{${\mathaccent"0362{#1}}^H$}% \setbox2\hbox{${\mathaccent"0362{\kern0pt#1}}^H$}% \ifdim\ht0=\ht2 #3\else #2\fi } %The bar will be moved to the right by a half of \macc@kerna, which is computed by amsmath: \newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna} %If there's a superscript following the bar, then no negative kern may follow the bar; %an additional {} makes sure that the superscript is high enough in this case: \newcommand*\widebar[1]{\@ifnextchar^{{\wide@bar{#1}{0}}}{\wide@bar{#1}{1}}} %Use a separate algorithm for single symbols: \newcommand*\wide@bar[2]{\if@single{#1}{\wide@bar@{#1}{#2}{1}}{\wide@bar@{#1}{#2}{2}}} \newcommand*\wide@bar@[3]{% \begingroup \def\mathaccent##1##2{% %Enable nesting of accents: \let\mathaccent\save@mathaccent %If there's more than a single symbol, use the first character instead (see below): \if#32 \let\macc@nucleus\first@char \fi %Determine the italic correction: \setbox\z@\hbox{$\macc@style{\macc@nucleus}_{}$}% \setbox\tw@\hbox{$\macc@style{\macc@nucleus}{}_{}$}% \dimen@\wd\tw@ \advance\dimen@-\wd\z@ %Now \dimen@ is the italic correction of the symbol. \divide\dimen@ 3 \@tempdima\wd\tw@ \advance\@tempdima-\scriptspace %Now \@tempdima is the width of the symbol. \divide\@tempdima 10 \advance\dimen@-\@tempdima %Now \dimen@ = (italic correction / 3) - (Breite / 10) \ifdim\dimen@>\z@ \dimen@0pt\fi %The bar will be shortened in the case \dimen@<0 ! 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We refine and extend our previous work [Int. Math. Res. Not. IMRN 2021, no. 12, 8845--8867] in which we settled the problem (for dual approximation) for hypersurfaces. We verify the GBSP for certain classes of nondegenerate submanifolds of codimension greater than $1$. Concretely, for codimension two or three, we provide examples of manifolds where the dependent variables can be chosen as quadratic forms. Our method requires the manifold to have even dimension at least the minimum of four and half the dimension of the ambient space. We conjecture that these restrictions on the dimension of the manifold are sufficient to provide similar examples in general. \end{abstract} \begin{altabstract} Le problème de Baker--Schmidt généralisé (1970) concerne la mesure de Hausdorff $f$ de l'ensemble des points $\Psi$-approximables sur une variété non dégénérée. Nous affinons et étendons notre travail précédent [Int. Math. Res. Not. IMRN 2021, no. 12, 8845-8867] dans lequel nous avons résolu le problème (pour l'approximation duale) pour les hypersurfaces. Nous vérifions le GBSP pour certaines classes de sous-variétés non dégénérées de codimension supérieure à 1. Concrètement, pour la codimension deux ou trois, nous donnons des exemples de variétés où les variables dépendantes peuvent être choisies comme des formes quadratiques. Notre méthode exige que la variété ait une dimension paire au moins égale au minimum de quatre et à la moitié de la dimension de l'espace ambiant. Nous conjecturons que ces restrictions sur la dimension de la variété sont suffisantes pour fournir des exemples similaires en général. \end{altabstract} \begin{document} \maketitle \section{Dual Diophantine approximation on manifolds} Let $n\geq 1$ be a fixed integer, $\qq:=(q_1,\ldots, q_n)\in \Z^n$, and $\xx=(x_1,\dots,x_n)\in\R^n$. Let $\Psi:\Z^n\to[0, \infty)$ be a \emph{multivariable approximating function}, that is, $\Psi$ has the property that $\Psi(\qq) \to 0 \text{ as } \|\qq\|:=\max(|q_1|, \ldots, |q_n|) \to \infty.$ Let $\theta$ be an arbitrary real number\footnote{ We remark that, in fact, our results still apply for any sufficiently smooth function $\theta(\xx)$, as in~\cite{HSS}, see Remark~\ref{fehlt} below.}. Consider the set \begin{equation*} \DDD:=\left\{\xx\in\R^n:\begin{array}{l} |\qq\cdot\xx+p+\theta|<\Psi(\qq) \ \text{for} \ {\rm i.m.} \ (p, \qq)\in\Z\times \Z^{n} \end{array} \right\}, \end{equation*} where ``i.m.'' stands for ``infinitely many''. A vector $\xx \in \R^n$ will be called \emph{$(\Psi, \theta)$-approximable} if it lies in the set $\DDD$. When $\theta= 0$, the problem reduces to the \emph{homogeneous} setting. We are interested in the ``size'' of the set $\DDD$ with respect to the $f$-dimensional Hausdorff measure $\HH^f$ for some dimension function $f$. By a dimension function $f$ we mean an increasing continuous function $f:\mathbb{R}\to \mathbb{R}$ with $f(0)=0$. Diophantine approximation on manifolds concerns the study of approximation properties of points in $\R^n$ which are functionally related or in other words restricted to a submanifold $\MM$ of $\R^n$. To estimate the size of sets of points $\xx\in\R^n$ which lie on a $k$-dimensional, non-degenerate\footnote{In this context ``non-degenerate'' means suitably curved, see~\cite{Beresnevich3,KleinbockMargulis2} for precise formulations}, analytic submanifold $\MM\subseteq \R^n$ is an intricate and challenging problem. The fundamental aim is to estimate the size of the set $\MM\cap\DDD$ in terms of Lebesgue measure, Hausdorff measure, and Hausdorff dimension. When asking such questions it is natural to phrase them in terms of a suitable measure supported on the manifold, since when $kbe; 2nd Then removed \begin{equation*} \HH^f( \DDD\cap\MM)=\begin{cases} 0 & \text{if } \sum\limits_{\qq\in\Z^n\setminus \{\0\}}\|\qq\|^k\Psi(\qq)^{1-k }f\left(\frac{\Psi(\qq)}{\|\qq\|}\right)< \infty;\\[3ex] \infty & \text{if } \sum\limits_{\qq\in\Z^n\setminus \{\0\}}\|\qq\|^k\Psi(\qq)^{1-k}f\left(\frac{\Psi(\qq)}{\|\qq\|}\right)=\infty. \end{cases} \end{equation*} % rewriting ! setminus 0 added \end{GBSP1} Note that $\HH^f$ is proportional to the standard Lebesgue measure when $f(r)=r^n$. In fact, the GBSP is stated in the most idealistic format and solving it in this form is extremely challenging. The main difficulties lie in the convergence case and therein constructing a suitable nice cover for the set $ \DDD\cap\MM$. Recently (2021), the authors with David Simmons settled the GBSP for hypersurfaces for both homogeneous and inhomogeneous settings with non-monotonic multivariable approximating functions~\cite{HSS}. We also proved several results for the one-dimensional manifolds such as non-degenerate planar curves or for Veronese curves in~\cite{HSS2} under some regularity conditions on the dimension function. In this paper we refine our framework set out in~\cite{HSS} and extend those results to certain classes of nondegenerate manifolds of higher codimension. We refer the reader to~\cite[Subsection~1.1]{HSS} for a description of the historical progression towards the GBSP. Specifically, we refer the reader to~\cite{BHH, BakerSchmidt, Huang, Huang2, Hussain, HSS2, KleinbockMargulis2}. For the recent developments of the GBSP for the simultaneous approximation, we refer the reader to~\cite[Remark~1.1]{HSS} or~\cite{Huang}. %analogue of this problem and developments in that direction \subsection{Setup and main result} \label{sandr} We first state some regularity conditions on the manifold $\MM$ and the dimension function $f$. Let $\MM$ be a manifold of dimension $n-l\ge 1$ in $\R^{n}$, with $n\geq 2$ and $l\geq 1$. Assume more precisely that $\MM$ is a graph of a $C^2$-map $g:U\to \R^{l}$, where $U\subset\R^{n-l}$ is a connected bounded open set. We denote this as $\MM = \Gamma(g)$. By default we will assume that vectors are line vectors, and use superscript $t$ for their transpose. Let $\MM$ be parametrised by % \begin{equation} \label{eq:via} \MM= \{ (x_{1},\ldots, x_{n-l}, g_{1}(\xx),\ldots, g_{l}(\xx)): \; \xx=(x_{1},\ldots,x_{n-l})\in U\subseteq \R^{n-l}\}. \end{equation} % Our conditions read as follows: \begin{enumerate}[label=(\Roman*)] \item\label{Sect1.1_I} %[(I)] Let $f$ be a dimension function satisfying \begin{equation} \label{fhypothesis} f(xy) \lesssim x^s f(y), \qquad \text{ for all } y < 1 < x \end{equation} for some $s < 2(n-l-1)$. \item\label{Sect1.1_II} %[(II)] The real symmetric $(n-l)\times (n-l)$ matrix $\Lambda=\Lambda(g, \boldsymbol{s},\xx)$ with entries \[ \Lambda_{j,i}= \sum_{u=1}^{l} s_u\cdot \frac{\partial^2g_u}{\partial x_i\partial x_j}(\xx), \qquad 1\le i,j\le n-l, \] for $g$ as above is regular for any choice of real $\boldsymbol{s}=(s_1, s_2,\ldots, s_l)\ne \0$ and all $\xx\in U\setminus S_{\MM}$ outside a set $S_{\MM}$ of $f$-measure $0$. \end{enumerate} While~\ref{Sect1.1_II} turns out to be rather complicated, we point out that~\ref{Sect1.1_I} holds as soon as the manifold has dimension at least three. This follows from the assumption that $r\to r^{-k} f(f)$ is decreasing, as in~\cite{HSS}, see paragraph~\ref{reh} below. Section~\ref{iandii} of this paper is reserved for a more detailed discussion on the conditions, followed by examples of manifolds satisfying the hypotheses in Section~\ref{examples}. Our new criterion for the convergence part of the GBSP to be proved in Section~\ref{proof} reads as follows. %we have that %\[ %P(\yy,\zz(\xx))\ne 0, \qquad \xx\in U, \; \yy\in \R^{l-1},\; \Vert %\yy\Vert \le 1. %\] %Hereby $\zz(\xx)$ is some explicit vector of length $\binom{l+1}{2}$ %derived in some way from partial derivatives of $g$ evaluated at %$\xx\in U$. %\comjohannes{At the moment the proof requires at most one solution %but the method should extend} \commumtaz{This may be regarded as the new definition of singularity??, i.e. whenever this property is not satisfied then we call such a manifold... } \begin{theo}\label{thm1} Let $\Psi$ be a multivariable approximating function. Let $f$ be a dimension function satisfying~\textup{\ref{Sect1.1_I}} and let $g: \R^{n-l}\to \R^{l}$ be a $C^2$-function satisfying~\textup{\ref{Sect1.1_II}}. Then if $\MM$ is given via $g$ by~\eqref{eq:via}, then \begin{equation*}\label{fw} \HH^f(\DDD\cap\MM)=0 \end{equation*} if the series \begin{equation*}\label{eqcon} \sum\limits_{\qq\in\Z^n\setminus \{\0\}}\|\qq\|^{n-l}\Psi(\qq)^{l+1-n}f\left(\frac{\Psi(\qq)}{\|\qq\|}\right) \end{equation*} converges. % rewriting ! CLAIM CORRECTED exponents were false n-> n-1, % and formulation added \end{theo} \begin{rema} \label{fehlt} It is possible to formulate a variant of Theorem~\ref{thm1} with a functional inhomogenity $\theta(\xx)$ as in~\cite{HSS}, however condition~\ref{Sect1.1_II} becomes more technical and less natural. \end{rema} \subsection{Corollaries (from a combination with the divergence results) }\label{sectionfibering} In this section, we detail some of the corollaries of our theorem along with some of the consequences. We begin by summarising the notation used. \smallskip \noindent{\bf Notation.} In the case where the dimension function is of the form $f(r):=r^s$ for some $s < k$, $\HH^f$ is simply denoted as $\HH^s$. On occasions we will consider functions of the form $\Psi(\qq)=\psi(\|\qq\|)$, and in this case we use $W_n^\theta(\psi)$ as a shorthand for $W_n^\theta(\Psi)$. The function $\psi:\R_{>0} \to \R_{>0}$ is called a \emph{single-variable approximating function}. $B_n(\xx, r)$ denotes a ball centred at the point $\xx\in\mathbb{R}^n$ of radius $r$. For real quantities $A,B$ and a parameter $t$, we write $A \lesssim_t B$ if $A \leq c(t) B$ for a constant $c(t) > 0$ that depends on $t$ only (while $A$ and $B$ may depend on other parameters). We write $A\asymp_{t} B$ if $A\lesssim_{t} B\lesssim_{t} A$. If the constant $c>0$ depends only on parameters that are constant throughout a proof, we simply write $A\lesssim B$ and $B\asymp A$. \smallskip The divergence part of the GBSP was proved by Badziahin--Beresnevich--Velani~\cite{BBV} for the $s$-dimensional Hausdorff measure\footnote{It is to be noted that the techniques used in proving~\cite[Theorem~2]{BBV} also work for the $f$-dimensional Hausdorff measure situation. } and for multivariable approximating functions satisfying a certain property $\bfP$ for any non-degenerate manifolds. Following the terminology of~\cite{BBV}, we say that an approximating function $\Psi$ satisfies property $\bfP$ if it is of the form $\Psi(\qq)=\psi(\|\qq\|_{\vv})$ for a monotonically decreasing function $\psi: \mathbb{R}_{>0}\to \mathbb{R}_{>0}$ (single-variable approximation function), $\vv=(v_{1},\ldots,v_n)$ with $v_i > 0$ and $\sum_{1\leq i\leq n} v_{i}=n$, and $\| \cdot \|_{\vv}$ defined as the quasi-norm $\|\qq\|_{\vv}= \max_i \vert q_{i}\vert^{1/v_{i}}$. When combined with Theorem~\ref{thm1} we obtain the following implication on the GBSP problem for non-degenerate (see~\cite{BBV}) manifolds for dimensions not less than $3$. \begin{corollary}\label{HSS:cor2} Let $\Psi$ be a decreasing %\comjohannes{ word added } multivariable approximating function satisfying property $\bfP$. Let $f$ be a dimension function satisfying~\textup{\ref{Sect1.1_I}} and let $g$ be a $C^2$-function satisfying~\textup{\ref{Sect1.1_II}}. Let $\MM$ be a non-degenerate manifold in $\mathbb{R}^{n}$ of dimension $n-l$, given via $g$ by~\eqref{eq:via}. Then \begin{equation*} \HH^f( \DDD\cap \MM)= \begin{cases} 0 & \textup{if } \sum\limits_{\qq\in\Z^n\setminus \{\0\}}\|\qq\|^{n-l}\Psi(\qq)^{l+1-n}f\left(\frac{\Psi(\qq)}{\|\qq\|}\right)< \infty;\\[3ex] \infty & \textup{if } \sum\limits_{\qq\in\Z^n\setminus \{\0\}}\|\qq\|^{n-l}\Psi(\qq)^{l+1-n}f\left(\frac{\Psi(\qq)}{\|\qq\|}\right)=\infty. \end{cases} \end{equation*} %where $\MM$ is the graph of $g$. \end{corollary} We emphasise again that only the divergence case, treated in~\cite{BBV}, assumes monotonicity on the approximating function $\Psi$. %As usual, %we write $\HH^s$ for $\HH^f$ when $f(x) = x^s$, $s>0$. \begin{corollary}\label{maincor} Let $\MM$ and $\Psi$ be as in Corollary~\ref{HSS:cor2} and let $s$ be a real number satisfying $s<2(n-l-1)$. Then \begin{equation*} \HH^s( \DDD\cap \MM)= \begin{cases} 0 & \textup{if } \sum\limits_{\qq\in\Z^n\setminus \{\0\}}\|\qq\|\left(\frac{\Psi(\qq)}{\|\qq\|}\right)^{s+l+1-n}< \infty;\\[3ex] \infty & \textup{if } \sum\limits_{\qq\in\Z^n\setminus \{\0\}}\|\qq\|\left(\frac{\Psi(\qq)}{\|\qq\|}\right)^{s+l+1-n}=\infty. \end{cases} \end{equation*} \end{corollary} %\comjohannes{ I am confused, where does $s>n-2$ come from? Nothing like that in HSS. If should be there then it should be explained why } For $l=1, n=2$ where $\MM$ is a planar curve, while the corollary and its proof are formally valid, the involved parameter range for $s$ is empty. We refer the reader to~\cite{HSS2} for the GBSP type results on non-degenerate curves such as Veronese curves, i.e. sets of the form $\{(x,x^{2},\ldots,x^n):x\in\R\}$. We remark that in view of Corollary~\ref{maincor}, Conjecture~\ref{C1} below would imply the GBSP for a large class of manifolds and any $\Psi$ with property $\mathbf{P}$. One of the consequences of Corollary~\ref{maincor} is the following Hausdorff dimension result. Let $\tau_{\Psi}$ be the lower order at infinity of $1/\Psi$, that is, \[ \tau_\Psi:=\liminf_{t\to\infty} \frac{\log(1/\Psi(t))}{\log t}, \ \ \text{where} \ \ \Psi(t)= \inf_{\xx\in\Z^n: \|\xx\|\le t} \Psi(\xx), \] %\comjohannes{ i think indeed we need $\le t$ in range } and we may assume $\tau_\Psi\ge n> 0$. Then from the definition of Hausdorff measure, Theorem~\ref{thm1} implies that for any approximating function $\Psi$ with lower order at infinity $\tau_{\Psi}$ and for $\MM$ as above (that is, with property~\ref{Sect1.1_II} of dimension $\dim\MM\geq 3$, we have \[ \dim_\HH (\WWW\cap\MM)\leq \dim \MM-1+\frac{n+1}{\tau_{\Psi}+1}. \] %\comjohannes{ do we need to further restrict $\tau$ for this? i think %its fine though } This Hausdorff dimension result was previously only known for either the planar curve~\cite{BHH, Hussain, Huang}, Veronese curve~\cite{Bernik}, or for the hypersurface~\cite{HSS}. \medskip \section{On conditions~\texorpdfstring{\ref{Sect1.1_I}}{(I)} and~\texorpdfstring{\ref{Sect1.1_II}}{(II)}}\label{iandii} \subsection{On Condition~\texorpdfstring{\upshape~\ref{Sect1.1_I}}{(I)}} \label{reh} Note that in the statement of the GBSP, the standard condition on the dimension function $f$ that $f(q)q^{-(n-l)}$ (recall $n-l=\dim \MM$) is decreasing is assumed. With this condition in hand, the condition~\ref{Sect1.1_I} is satisfied as soon as \[ n-l\geq 3. \] From the aforementioned decay property we see that $f(xy)\le x^{n-l}f(y)$ for $y<1l\ge 1$ a {\em good pair} if~\textup{\ref{Sect1.1_II}} holds for some $g:\R^{n-l}\to \R^l$ as in the introduction. \end{defi} By the above observation, the induced $g$ of any good pair can be taken a quadratic form defined globally. It is obvious that any pair $(1,n)$ is good. For larger $l$, the following general going-up and going-down properties for good pairs are straightforward. \begin{prop} \label{prop} If $(l,n)$ is a good pair, then so is \begin{enumerate}[label=(\roman*)] \item\label{prop6_i} %[\rm (i)] the pair $(l-t,n-t)$ for any integer $0\le t\le l-1$. \item\label{prop6_ii} %[\rm (ii)] the pair $(l,\tilde{n})$ with $\tilde{n}=n+t(n-l)$ %\[ %\tilde{g}=(g_1,\ldots,g_{l},g_1,\ldots,g_{l},\ldots,g_1,\ldots,g_{%l}) %\] % where $g=(g_1,\ldots,g_{l})$ is repeated $t$ times, for any integer $t\ge 0$. \end{enumerate} \end{prop} Claim~\ref{prop6_i} can be seen by specialisation of $t$ variables, for example via putting $s_1~=~\cdots=~s_{t+1}$. The proof of~\ref{prop6_ii} will become apparent from the examples in Section~\ref{examples}. On the other hand, it is in general unclear if $(l,n)$ being a good pair will imply the same for $(l,\tilde{n})$ with a general larger $\tilde{n}>n$, even if $\tilde{n}-l$ is even (see Obstruction~1 below). On the other hand, condition~\ref{Sect1.1_II} has some natural limitations, as captured in the following obstructions. \setcounter{cdrthm}{0} \goodbreak \begin{Obstruction}\label{obst1} %\noindent $\textbf{Obstruction 1:}$ %\label{rem} If $l\ge 2$ and $n-l$ is odd, then $(l,n)$ is not a good pair, that is, the condition~\textup{\ref{Sect1.1_II}} cannot hold. While a short proof seems to follow directly from the positive answer to Hilbert's XVII'th problem, we want to explicitly explain how this special case can be handled. First assume $l=2$. Indeed, if either the coefficient of $x_1^{n-l}$ or $x_2^{n-l}$ of the polynomial $P(x_1,x_2)$ defined above vanishes, then we may take $x_1$ arbitrary and $x_2=0$ or vice versa, hence we get non-trivial solutions for $\det \Lambda=P(x_1,x_2)=0$. If otherwise both coefficients are not $0$, then by choosing any $x_2\ne 0$ we get a single-variable polynomial in $x_1$ of odd degree with non-zero constant term, again inducing a non-trivial solution for $\det \Lambda=P(x_1,x_2)=0$. Finally if $l>2$, by specialising $l-2$ variables $x_3=\cdots=x_l=0$, we get a polynomial $Q(x_1,x_2)$ in two variables. Regardless if $Q\equiv 0$ is the constant $0$ polynomial or not, by the observations for $l=2$ above, we may choose at least one of $x_1, x_2$ (almost) arbitrary for a solution of $\det \Lambda=0$. Thus we again find a non-trivial solution for $\det \Lambda=0$ in $s_1,\ldots,s_l$. %\end{rema} \end{Obstruction} \begin{Obstruction}\label{obst2} %\noindent $\textbf{Obstruction 2:}$ We require $n\ge 2l$ for $(l,n)$ being a good pair. Otherwise if $n-l2$}{l>2}} Now let us consider $l>2$. Together with the restrictions \[ n-l\equiv 0\bmod 2,\qquad n-l\ge 3,\qquad n\ge 2l, \] from Section~\ref{reh} and Obstructions~\ref{obst1} and~\ref{obst2}, the easiest example is $l=3, n=7$. The following general criterion for~\ref{Sect1.1_II}, or good pairs, involving ``definite determinants'' essentially comes from specialising certain variables (second order derivatives of $g$). \begin{lemma} \label{laura} Let $n> l\ge 1$ be integers. Assume there exists a symmetric $(n-l)\times (n-l)$ matrix $M_{l,n}$ with the entries \[ M_{l,n}(i,j)\in \{ 0, \pm z_1 , \ldots, \pm z_l \}, \qquad 1\le i,j\le n-l, \] for formal variables $z_v$, $1\le v\le l$, so that $\det M_{l,n} \ne 0$ for any choice of real $z_v$ not all $0$. Then $(l,n)$ is a good pair. \end{lemma} Hence the existence of $M_{l,n}$ as in the lemma implies that there exist $(n-l)$-dimensional submanifolds of $\R^n$, defined as in~\eqref{eq:via} via $g=(g_1,\ldots,g_{l})$ with $g_j(\xx)$ real quadratic forms, that satisfy~\ref{Sect1.1_II} on $U=\R^{n-l}$. Again, the hypothesis of Lemma~\ref{laura} forces $n\ge 2l$ and $n-l$ to be even. \begin{rema} More generally, instead of $M_{l,n}(i,j)=\pm z_v$ or $0$, we may take the matrix entries arbitrary linear combinations of the $z_v$, for the same conclusion. This generalised condition is in fact equivalent to~\ref{Sect1.1_II}. The proof is essentially the same as below. \end{rema} %The claim becomes stronger the larger %$l$ is compared to $n$, as we can also decrease $l$ by choosing %certain variables equal. \begin{proof}[Proof of Lemma~\ref{laura}] Choose any set of $l$ linearly independent (over $\R$) vectors $L_v=(a_{1,v},\ldots,a_{l,v})$, $1\le v\le l$, in $\R^l$ and define the linear forms in $l$ variables $s_1,\ldots,s_l$ \[ L_{v}\cdot \boldsymbol{s}= a_{1,v}s_1+\cdots + a_{l,v}s_l, \quad 1\leq v \le l. \] Given $M_{l,n}$ as in the lemma, we construct a matrix $M_{l,n}^{\prime}$ with entries in formal variables $s_1,\ldots,s_l$ as follows: We identify $z_v$ with $L_v\cdot \boldsymbol{s}$, that is, if for a pair $(i,j)$ the index $v=v(i,j)$ is so that $M_{l,n}(i,j)=\pm z_v$, then we choose the entry at position $(i,j)$ of $M_{l,n}^{\prime}$ equal to $M_{l,n}^{\prime}(i,j)=\pm L_v\cdot \boldsymbol{s}$, with the same sign choice as above. Else if $M_{l,n}(i,j)=0$ then keep the value $M_{l,n}^{\prime}(i,j)=0$. Then $M_{l,n}^{\prime}$ is well-defined and depends on $s_1,\ldots,s_l$. By linear independence of the $L_v$ and the hypothesis of the lemma, for any non-trivial choice of real numbers $s_1, \ldots,s_l$, the matrix $M_{l,n}^{\prime}$ has non-zero determinant. We finally notice that there is a one-to-one correspondence between any such collection of coefficient vectors $L_{v}$ and a collection of quadratic forms $g_1,\ldots, g_l$ via identifying for $v=v(i,j)$ as above $a_{u,v}=\partial^2 g_u/\partial x_i\partial x_j$ for $1\le u\le l$. In other words, $a_{u,v}x_ix_j$ if $i\ne j$ and $(a_{u,v}/2) x_i^2$ if $i=j$ is the term containing $x_ix_j$ resp. $x_i^2$ in the quadratic form $g_u(\xx)=\sum_{i,j} a_{u,v}x_ix_j$. Hence~\ref{Sect1.1_II} holds for the globally defined function $g=(g_1,\ldots,g_l)$, so $(l,n)$ is a good pair. \end{proof} In the proof, we may just let $L_v$ the canonical base vector in $\R^l$ for $1\le v\le l$ so that essentially $z_v$ equals $s_v$, however the proof contains more information. Hence the problem~\ref{Sect1.1_II} can be relaxed to finding suitable matrices $M_{l,n}$ as in the lemma. For $l=2, n=4$ we can take the matrix \[ M_{2,4}=\begin{pmatrix} z_1 & z_2 \\ z_2 & -z_1 \end{pmatrix} \] which leads to Example~\ref{Ex1}. More generally, for $l=2$ and $n=2v$ with $v\ge 2$ an even integer, we may take a matrix consisting of $v-1$ copies of $2\times 2$ ``diagonal'' blocks as above and zeros elsewhere, similar to Example~\ref{e2} and Proposition~\ref{prop}. For $l=3, n=7$ we can take \begin{equation} \label{eq:37} M_{3,7}=\begin{pmatrix} z_1 & z_2 & z_3 & -z_{3} \\ z_2 & -z_1 & z_3 & z_3 \\ z_3 & z_3 & z_1 & z_2 \\ -z_3 & z_3 & z_2 & -z_1 \end{pmatrix} \end{equation} which has determinant $\det(M_{3,7})= (z_1^2+z_2^2)^2 + 4z_3^4$. Our construction leads to the following example. \begin{exam} Let $l=3, n=7$. Taking the linear forms $L_{v}$ from the proof of Lemma~\ref{laura} the canonical base vectors $(1,0,0), (0,1,0), (0,0,1)$ of $\R^3$ and inserting for the $z_v$ in $M_{3,7}$ from~\eqref{eq:37}, leads to $g=(g_1,g_2,g_3)$ with quadratic form entries $g_u(\xx)=g_u(x_1,x_2,x_3,x_4)$ given by \begin{align*} &g_1(\xx)=\frac{x_1^2-x_2^2+x_3^2-x_4^2}{2}, \\ &g_2(\xx)= x_1x_2+x_3x_4,\\ &g_3(\xx)= x_1x_3+x_2x_3+x_2x_4-x_1x_4. \end{align*} So, the manifold becomes \[ \mathcal{M}= \{ (x_1,x_2,x_3,x_4, g_1(\xx), g_2(\xx), g_3(\xx): x_i\in\R \}, \] which satisfies~\ref{Sect1.1_I} and~\ref{Sect1.1_II}, and thus GBSP for any decreasing $\Psi$ with property $\mathbf{P}$. \end{exam} For $l=3$ we may take any $n\in\{7,11,15,19,\ldots\}$ again by repeating this $4\times 4$-block matrix $M_{3,7}$ along the ``diagonal''. It is unclear if $l=3$ and $n\in\{ 9, 13, 17, \ldots \}$ can be achieved. It would suffice to verify this for $n=9$ to infer the claim for all $n$ in the list by considering matrices decomposing into two types of diagonal blocks, $M_{3,7}$ and the vacant $M_{3,9}$. This would confirm Conjecture~\ref{C1} for $l=3$ as well. Unfortunately, we are unable to find a suitable matrix $M_{3,9}$. %\comjohannes{ would be great to find it (if exists)!!!! } The above discussion on Lemma~\ref{laura} motivates the following problem implicitly stated within Conjecture~\ref{C1}. \begin{problem} Given $l\ge 3$, what is the minimum $n$ so that a matrix $M_{l,n}$ as in Lemma~\ref{laura} exists? Equivalently, given even $n-l$, what is the largest $l$ for which the hypothesis holds for some matrix. \end{problem} It is unclear if such $n$ exists at all if $l\ge 4$. As remarked in Section~\ref{condii}, in all examples of Section~\ref{examples}, we can manipulate the manifold by adding any functions with uniformly (in absolute value) small enough second order derivatives to the $l$ functionally dependent variables. In particular, for any analytic functions defined on a neighbourhood of $\0\in\R^{n-l}$ with quadratic terms as in our examples above, GBSP holds upon possibly shrinking the neighbourhood. \section{Proof of Theorem \ref{thm1}}\label{proof} Let us first clarify some notation. With $g$ as above, write \[ g(\xx)=(g_{1}(\xx),\ldots, g_{l}(\xx))\in \R^{l}, \qquad \xx\in U, \] % which we will assume as a row vector. Furthermore, write $\nabla g^{t}$ for the $(n-l)\times l$ matrix with $i$-th partial derivative vector in $\R^l$ of the transpose of $g$ (thus a column vector) in the $i$-th column ($1\leq i\le n-l$). Denote by $\nabla^2 g_u^{t}\in \mathbb{R}^{(n-l)\times (n-l)}$ the Hessians of the (transposed) coordinate functions $g_u^{t}$, $1\le u\le l$, where again the $i$-th column consists of the partial derivative vector $\partial g_u/\partial x_i$, $1\le i\le n-l$. The proof is a refinement and extension of arguments presented in our previous paper~\cite{HSS}. Here we only detail some modifications and other necessary details. For any $\qq\in \Z^n$ and $p\in\Z$, analogously to~\cite{HSS} we define \[ S(\qq,p) = S_{\Psi,\theta}(\qq,p) = \{\xx\in K : |\qq\cdot (\xx,g(\xx))^{t} - p - \theta| < \Psi(\qq)\}, \] % rewriting ! wording changed where $K$ is a compact subset of $U$. However, notice that our $g(\xx)$ is a vector here. Write $\qq = \tilde{q}\cdot(\rr,\boldsymbol{s})$ for \[ \tilde{q}=\max_{n-l+1\leq i\leq n}\vert q_{i}\vert \] for $\rr\in\Q^{n-l}$ and $\boldsymbol{s}\in\Q^{l}$. % rewriting ! n -> n-1 : R -> Q Let $a = a(\qq)= (p+\theta)/\tilde{q}$ and $\rho = \Psi(\qq)/|\tilde{q}|$ if $\tilde{q}\ne 0$ (assume this for now). Then $\|\boldsymbol{s}\| =1$, and $\tilde{q} \le \|\qq\|$ by definition. For any $\qq\in \Z^n$ and $p\in\Z$, we will bound the size of this set which is equivalently given as \[ S(\qq,p) = \{\xx\in K : |\rr\cdot\xx^{t} + \boldsymbol{s}\cdot g(\xx)^{t} - a| < \rho\}. \] Similarly as in~\cite{HSS}, for fixed $p$ and $\qq$, % rewriting ! added define a function $h: \R^{n-l}\to \R$ by \[ h(\xx) = \rr\cdot\xx^{t} + \boldsymbol{s}\cdot g(\xx)^{t} - a, \] for $\rr, \boldsymbol{s}, a$ induced by $\qq, p$ as above. % rewriting ! added Finally in case $\tilde{q}=0$, we instead let \[ \rr=(q_1,\ldots,q_{n-l}), \qquad h(\xx)= \rr\cdot \xx^{t} - p - \theta, \qquad \rho=\Psi(\qq). \] As in~\cite{HSS} we see that in either case \[ S(\qq,p)= \{ \xx\in K: |h(\xx)|<\rho \}. \] We now identify $\nabla^2 h$ with the matrix $\Lambda$ of the theorem by the following calculation: For $1\le i\le n-l$, the $i$-th entry of the vector $\nabla h(\xx)$ equals $r_i+ \boldsymbol{s}\cdot G_i(\xx)^t$ with \[ G_i(\xx)^{t}= (\partial g^{t}/\partial x_i)(\xx)= (\partial g_1/\partial x_i(\xx), \ldots, \partial g_l/\partial x_i(\xx))^{t} \] the $i$-th column of $\nabla g^{t}$ evaluated at $\xx$ if $\tilde{q}\ne 0$, and $\nabla h(\xx)_i=r_i$ if $\tilde{q}=0$. In the sequel we assume $\tilde{q}\ne 0$, else similarly the argument is analogous to~\cite{HSS}. So this $i$-th entry of $\nabla h(\xx)$, denote it by $\nabla h(\xx)_i$, reads \[ \nabla h(\xx)_i = r_i + \boldsymbol{s}\cdot \left(\frac{\partial g}{\partial x_i}(\xx)\right)^t= r_i + \sum_{u=1}^{l} s_u\cdot (\partial g_u/\partial x_i(\xx)), \qquad 1\le i\le n-l. \] Then \[ \nabla h(\xx)- \sum_{i=1}^{n-l} r_i \ee_i= \sum_{i=1}^{n-l} \boldsymbol{s}\cdot \left(\frac{\partial g}{\partial x_i}(\xx)\right)^t\cdot \ee_i= \sum_{i=1}^{n-l}\sum_{u=1}^{l} s_u\cdot (\partial g_u/\partial x_i(\xx))\cdot \ee_i, \] with $\ee_i$ the canonical base vectors in $\R^{n-l}$. Hence the quadratic matrix $\nabla^2 h(\xx)$ with $j$-th line $\partial\nabla h(\xx)/\partial x_j$ has entries \begin{equation} \label{eq:NEUE} \nabla^2 h(\xx)_{j,i}= \sum_{u=1}^{l} s_u\cdot \frac{\partial^2g_u}{\partial x_i\partial x_j}(\xx), \qquad 1\le i,j\le n-l, \end{equation} (and vanishes if $\tilde{q}=0$). Thus indeed we may identify $\nabla^2 h$ with $\Lambda$. In contrast to~\cite{HSS}, the matrix $\nabla^2 h(\xx)$ now also depends on $\qq\in\Z^n$ via its dependence on the induced rational vector $\boldsymbol{s}$, however by a compactness argument we will deal with this issue. By assumption of~\ref{Sect1.1_II}, for any unit vector $\boldsymbol{s}\in\R^l$ and any $\xx\in K$, the determinant of $\nabla^2 h(\xx)$ does not vanish. For simplicity assume for the moment $\qq$ and thus $\boldsymbol{s}$ are fixed. %By compactness %of the set $\{ \boldsymbol{s}\in\R^{l}: \Vert\boldsymbol{s}\Vert=1\}$ %and the continuous dependency of $\nabla^2 h(\xx)$ on %$\boldsymbol{s}$, Then as in~\cite{HSS}, for some $\varepsilon>0$, the matrices $\nabla^2 h(\xx)$ with $\xx\in K$ belong to a compact, convex set of matrices (identified with $\R^{(n-l)\times (n-l)}$) with determinants at least $\varepsilon$. Now the same argument as in~\cite{HSS} based on the mean value inequality gives the analogue of~\cite[Claim~2.3]{HSS}, which reads: \begin{enonce*}[remark]{Claim} If the norm of $\qq\in\Z^n$ is large enough, then either there exists $\vv\in\R^{n-l}$ such that \[ \Vert \nabla h(\xx)\Vert \asymp \Vert \xx-\vv\Vert, \qquad \xx\in K, \] or \[ \Vert \nabla h(\xx)\Vert \asymp \Vert(\rr,\boldsymbol{s})\Vert \asymp \Vert (\rr,1)\Vert, \qquad \xx\in K. \] Either way, we have $\Vert \nabla h(\xx)\Vert\ll 1$ for all $\xx\in K$. We remark that the last claim is obvious by $\Vert \boldsymbol{s}\Vert=1$ and \eqref{eq:NEUE}. It should be pointed out that following the proof in~\cite{HSS}, the implied constants in the lemma will still depend on $\qq$ in the form of the dependence on $\boldsymbol{s}$. However, by the continuous dependence of $\nabla^2 h(\xx)$ on $\boldsymbol{s}$ and the compactness of the set $\{ \boldsymbol{s}\in\R^{l}: \Vert\boldsymbol{s}\Vert=1\}$, it is easily seen that we can find uniform constants. The remainder of the proof of Theorem~\ref{thm1} works analogously as in~\cite{HSS}, where we consider two cases and we replace $n$ by $n-l+1$ consistently and omit $\nabla \theta(\xx)$ and $\nabla^2 \theta(\xx)$, as we consider $\theta$ constant. It is worth noticing that in Case 1 we apply the analogue of~\cite[Lemma~2.4]{HSS}, which we again want to state explicitly for convenience: \end{enonce*} \begin{lemma}[Hussain, Schleischitz, Simmons] \label{lemmacover} Assume $n,l$ are positive integers satisfying $n>l+1$. %\comjohannes{ i think we need this not n greater two times l} Let $\phi: U \subset \R^{n-l}\to \R$ be a $C^2$ function. Fix $\alpha>0$, $\delta>0$, and $\xx\in U$ such that $B_{n-l}(\xx,\alpha) \subset U$. There exists a constant $C > 0$ depending only on $n$ such that if \begin{equation}\label{ine} \|\nabla \phi(\xx)\| \geq C \alpha \sup_{\zz\in U} \|\nabla^2 \phi(\zz)\|, \end{equation} then the set $$S(\phi,\delta) = \{\yy\in B_{n-l}(\xx,\alpha) : |\phi(\yy)| < \|\nabla \phi(\xx)\| \delta\}$$ can be covered by $\asymp (\alpha/\delta)^{n-l-1}$ balls of radius $\delta$. \end{lemma} Indeed, this is precisely~\cite[Lemma~2.4]{HSS}, upon replacing $n$ by $n-l+1$. Rest of the proof of Theorem follows just like~\cite{HSS} with obvious adaptations mentioned above. Specifically, by taking into account the observation that the implied constants in the Claim above are independent of $p, \qq$ (or $\boldsymbol{s}$). That ultimately leads to the sufficient condition $s<2(n-l-1)$ in place of $s<2n-2$ from~\cite{HSS}, which agrees with condition~\ref{Sect1.1_I}. \section*{Acknowledgments} We are thankful to the referee for some useful comments. \section*{Declaration of interests} The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations. \printbibliography \end{document}