Singularity of sparse Bernoulli matrices

The Littlewood–Offord theory — the study of anti-concentration properties of random variables — has been a crucial ingredient of many recent results on invertibility of random matrices, starting with the work of Tao–Vu [47]. In particular, the breakthrough result [41] of Rudelson–Vershynin mentioned in the introduction, is largely based on studying the Lévy function $\mathcal{Q}(\langle{\bf C}_{1}(A),{\bf Y}\rangle,t)$, with ${\bf C}_{1}(A)$ being the first column of the random matrix $A$ and ${\bf Y}$ — a random unit vector orthogonal to the remaining columns of $A$.

We recall that given a random vector $X$ taking values in ${\mathbb{R}}^{n}$, the Lévy concentration function $\mathcal{Q}(X,t)$ is defined by

$$\mathcal{Q}(X,t):=\sup\limits_{y\in{\mathbb{R}}^{n}}{\mathbb{P}}\big{\{}\|X-y\|\leq t\big{\}},\quad t\geq 0;$$

in particular for a scalar random variable $\xi$ we have $\mathcal{Q}(\xi,t):=\sup\limits_{\lambda\in{\mathbb{R}}}{\mathbb{P}}\{|\xi-\lambda|\leq t\}$. A common approach is to determine structural properties of a fixed vector which would imply desired upper bounds on the Lévy function of its scalar product with a random vector (say, a matrix’ column). The classical result of Erdős–Littlewood–Offord [12, 21] asserts that whenever $X$ is a vector in ${\mathbb{R}}^{n}$ with i.i.d. $\pm 1$ components, and $y=(y_{1},\dots,y_{n})\in{\mathbb{R}}^{n}$ is such that $|y_{i}|\geq 1$ for all $i$, we have

$$\mathcal{Q}(\langle X,y\rangle,t)\leq Ct\,n^{-1/2}+Cn^{-1/2},$$

where $C>0$ is a universal constant. It can be further deduced from the Lévy–Kolmogorov–Rogozin inequality [38] that the above assertion remains true whenever $X$ is a random vector with independent components $X_{i}$ satisfying $\mathcal{Q}(X_{i},c)\leq 1-c$ for some constant $c>0$. More delicate structural properties, based on whether components of $y$ can be embedded into a generalized arithmetic progression with prescribed parameters were employed in [47] to prove superpolynomially small upper bounds on the singularity probability of discrete random matrices.

The Least Common Denominator (LCD) of a unit vector introduced in [41] played a central role in establishing the exponential upper bounds on the matrix singularity under more general assumptions on the entries’ distributions. We recall that the LCD of a unit vector $y$ in ${\mathbb{R}}^{n}$ can be defined as

$${\rm LCD}(y):=\inf\big{\{}\theta>0:\;{\rm dist}(\theta y,{\mathbb{Z}}^{n})\leq\min(c_{1}\|\theta y\|,c_{2}\sqrt{n})\big{\}}$$

(5)

for some parameters $c_{1},c_{2}\in(0,1)$. The small ball probability theorem of Rudelson and Vershynin [41] states that given a vector $X$ with i.i.d. components of zero mean and unit variance satisfying some additional mild assumptions,

$$\mathcal{Q}(\langle X,y\rangle,t)\leq Ct+\frac{C^{\prime}}{{\rm LCD}(y)}+2e^{-c^{\prime}n}$$

for some constants $C,C^{\prime},c^{\prime}>0$ (see [42] for a generalization of the statement). The LCD, or its relatives, were subsequently used in studying invertibility of non-Hermitian square matrices under broader assumptions [37, 30, 31], and delocalization of eigenvectors of non-Hermitian random matrices [43, 34, 32], among many other works.

Anti-concentration properties of random linear combinations naturally play a central role in the current work, however, the measures of unstructuredness of vectors existing in the literature do not allow to obtain the precise estimates we are aiming for. Here, we develop a new functional for dealing with linear combinations of dependent Bernoulli variables.

Given $n\in{\mathbb{N}}$, $1\leq m\leq n/2$, a vector $y\in{\mathbb{R}}^{n}$ and parameters $K_{1},K_{2}\geq 1$, we define the degree of unstructuredness (u-degree) of vector $y$ by

$\displaystyle{\bf UD}_{n}(y,m,K_{1},K_{2}):=\sup\bigg{\{}$

$\displaystyle t>0:\;A_{nm}\sum\limits_{S_{1},\dots,S_{m}}\;\int\limits_{-t}^{t}\prod\limits_{i=1}^{m}\psi_{K_{2}}\big{(}\big{|}{\mathbb{E}}\exp\big{(}2\pi{\bf i}\,y_{\eta[S_{i}]}\,m^{-1/2}s\big{)}\big{|}\big{)}\,ds\leq K_{1}\bigg{\}},$

(6)

where the sum is taken over all sequences $(S_{i})_{i=1}^{m}$ of disjoint subsets $S_{1},\dots,S_{m}\subset[n]$, each of cardinality $\lfloor n/m\rfloor$ and

$\displaystyle A_{nm}=\frac{\big{(}(\lfloor n/m\rfloor)!\big{)}^{m}\,(n-m\lfloor n/m\rfloor)!}{n!}\cdot$

(7)

Here $\eta[S_{i}]$, $i\leq m$, denote mutually independent integer random variables uniformly distributed on respective $S_{i}$’s. The function $\psi_{K_{2}}$ in the definition acts as a smoothing of $\max(\frac{1}{K_{2}},t)$, with $\psi_{K_{2}}(t)=\frac{1}{K_{2}}$ for all $t\leq\frac{1}{2K_{2}}$ and $\psi_{K_{2}}(t)=t$ for all $t\geq\frac{1}{K_{2}}$ (we prefer to skip discussion of this purely technical element of the proof in this section, and refer to the beginning of Section 4 for the full list of conditions imposed on $\psi_{K_{2}}$).

The functional ${\bf UD}_{n}(y,m,K_{1},K_{2})$ can be understood as follows. The expression inside the supremum is the average value of the integral

$$\int\limits_{-t}^{t}\prod\limits_{i=1}^{m}\psi_{K_{2}}\big{(}\big{|}{\mathbb{E}}\exp\big{(}2\pi{\bf i}\,y_{\eta[S_{i}]}\,m^{-1/2}s\big{)}\big{|}\big{)}\,ds,$$

with the average taken over all choices of sequences $(S_{i})_{i=1}^{m}$. The function under the integral, disregarding the smoothing $\psi_{K_{2}}$, is the absolute value of the characteristic function of the random variable $\langle y,Z\rangle$, where $Z$ is a random $0/1$–vector with exactly $m$ ones, and with the $i$-th one distributed uniformly on $S_{i}$. A relation between the magnitude of the characteristic function and anti-concentration properties of a random variable (the Esseen lemma (Lemma 3.12 below) has been commonly used in works on the matrix invertibility (see, for example, [40]), and determines the shape of the functional ${\bf UD}_{n}(\cdot)$. The definition of the u-degree is designed specifically to work with random $0/1$–vectors having a fixed sum (equal to $m$). The next statement follows from the definition of ${\bf UD}_{n}(\cdot)$ and the Esseen lemma.

The principal difference of the u-degree and the above theorem from the notion of the LCD and (5) is that the former allow to obtain stronger anti-concentration inequalities in the same regime of sparsity, assuming that the coefficient vector $y$ is sufficiently unstructured. In fact, under certain conditions, sparse random $0/1$ vectors with prescribed support cardinality admit stronger anti-concentration inequalities compared to the i.i.d. model.

The last principle can be illustrated by taking the coefficient vector $y$ as a “typical” vector on the sphere $S^{n-1}$. First, assume that $b_{1},\dots,b_{n}$ are i.i.d. Bernoulli($p$) , with $p<1/2$. Then it is easy to see that for almost all (with respect to normalized Lebesgue measure) vectors $y\in S^{n-1}$,

$$\mathcal{Q}\Big{(}\sum_{i=1}^{n}y_{i}b_{i},0\Big{)}=(1-p)^{n}.$$

In words, for a typical coefficient vector $y$ on the sphere, the linear combination $\sum_{i=1}^{n}y_{i}b_{i}$ takes distinct values for any two distinct realizations of $(b_{1},\dots,b_{n})$, and thus the Lévy function at zero is equal to the probability measure of the largest atom of the distribution of $\sum_{i=1}^{n}y_{i}b_{i}$ which corresponds to all $b_{i}$ equal to zero. In contrast, if the vector $(b_{1},\dots,b_{n})$ is uniformly distributed on the set of $0/1$–vectors with support of size $d=pn$, then for almost all $y\in S^{n-1}$, the random sum $\sum_{i=1}^{n}y_{i}b_{i}$ takes ${n\choose d}$ distinct values. Thus,

$$\mathcal{Q}\Big{(}\sum_{i=1}^{n}v_{i}b_{i},0\Big{)}={n\choose np}^{-1},$$

where ${n\choose np}^{-1}\ll(1-p)^{n}$ for small $p$.

The above example provides only qualitative estimates and does not give an information on the location of the atoms of the distribution of $\sum_{i=1}^{n}y_{i}b_{i}$. The notion of the u-degree addresses this problem. The following theorem, which is the main result of Section 4, asserts that with a very large probability the normal vector to the (say, last) $n-1$ columns of our matrix $M_{n}$ is either very structured or has a very large u-degree, much greater than the critical value $(1-p)^{-n}$.

We would like to emphasize that the parameter $s$ in this theorem can take values less than one, in the regime when the matrix $M_{n}$ typically has null rows and columns. In this respect, the restriction $p\geq C\ln n/n$ in the main theorem comes from the treatment of structured vectors.

The proof of Theorem 2.2 is rather involved, and is based on a double counting argument and specially constructed lattice approximations of the normal vectors. We refer to Section 4 for details. Here, we only note that, by taking $R$ as a sufficiently large constant, the theorem implies the relation (4), hence, accomplishes the treatment of unstructured vectors.

In this subsection we discuss our treatment of the set of structured vectors, ${\mathcal{S}}_{n}$. In the proof we partition the set ${\mathcal{S}}_{n}$ into several subsets and work with them separately. In a simplistic form, the structured vectors are dealt with in two ways: either by constructing discretizations and taking the union bound (variations of the $\varepsilon$–net argument), or via deterministic estimates in the case when there are very few very large components in the vector. We note here that the discretization procedure has to take into account the non-centeredness of our random matrix model: while in case of centered matrices with i.i.d. components (and under appropriate moment conditions) the norm of the matrix is typically of order $\sqrt{n}$ times the standard deviation of an entry, for our Bernoulli($p$) model it has order $pn$ (i.e., roughly $\sqrt{p}n$ times the standard deviation of an entry), which makes a direct application of the $\varepsilon$–net argument impossible. Fortunately, this large norm is attained only in one direction — the direction of the vector ${\bf 1}=(1,1,\dots,1)$ while on the orthogonal complement of ${\bf 1}$ the typical norm is $\sqrt{pn}$. Therefore it is enough to take a standard net in the Euclidean norm and to make it denser in that one direction, which almost does not affect the cardinality of the net. We refer to Section 3.6 for details.

Let us first describe our approach in the (simpler) case when $p\in(q,c)$, where $c$ is a small enough absolute constant and $q\in(0,c)$ is a fixed parameter (independent of $n$). We introduce four auxiliary sets and show that the set of unit structured vectors, ${\mathcal{S}}_{n}\cap S^{n-1}$, is contained in the closure of their union.

The first set, $\mathcal{B}_{1}$, consists of unit vectors close to vectors of the canonical basis, specifically, unit vectors $x$ satisfying $x_{1}^{*}>6pnx_{2}^{*}$, where $x^{*}$ denotes the non-inreasing rearrangement of the vector $(|x_{i}|)_{i\leq n}$. For any such vector $x$ the individual bound is rather straightforward — conditioned on the event that there are no zero columns in our matrix $M$, and that the Euclidean norms of the matrix rows are not too large, we get $Mx\neq 0$. This class is the main contributor to the bound $(1+o_{n}(1))n(1-p)^{n}$ for non-invertibility over the structured vectors ${\mathcal{S}}_{n}$.

For the other three sets we use anti-concentration probability estimates and discretizations. An application of Rogozin’s lemma (Proposition 3.9) implies that probability of having small inner product of a given row of our matrix with $x$ is small, provided that there is a subset $A\subset[n]$ such that the maximal coordinate of $P_{A}x$ is bounded above by $c\sqrt{p}\|P_{A}x\|$, where $\|\cdot\|$ denotes the standard Euclidean norm and $P_{A}$ is the coordinate projection onto ${\mathbb{R}}^{A}$. Combined with tensorization Lemma 3.8 this implies exponentially (in $n$) small probability of the event that $\|Mx\|$ is close to zero — see Proposition 3.10 below. Specifically, we define $\mathcal{B}_{2}$ as the set of unit vectors satisfying the above condition with $A=[n]$, that is, satisfying $x_{1}^{*}\leq c\sqrt{p}$, and for $\mathcal{B}_{3}$ we take all unit vectors satisfying the condition with $A=\sigma_{x}([2,n])$, that is, satisfying $x_{2}^{*}\leq c\sqrt{p}\|P_{\sigma_{x}([2,n])}x\|$, where $\sigma_{x}$ is a permutation satisfying $x^{*}_{i}=|x_{\sigma_{x}(i)}|$, $i\leq n$. For vectors from these two sets we have very good individual probability estimates, but, unfortunately, the complexity of both sets is large — they don’t admit nets of small cardinality. To overcome this issue, we have to redefine these sets by intersecting them with specially chosen sets of vectors having many almost equal coordinates. For the precise definition of such sets, denoted by $U(m,\gamma)$, see Subsection 3.6. A set $U(m,\gamma)$ is a variant of the class of almost constant vectors, $\mathcal{AC}(\rho)$ (see (9) below), introduced to deal with general $p$. Having a large part of coordinates of a vector almost equal to each other reduces the complexity of the set making possible to construct a net of small cardinality. This resolves the problem and allows us to deal with these two classes of sets. The remaining class of vectors, $\mathcal{B}_{4}$, consists of vectors $x$ with $x_{1}^{*}\geq x_{2}^{*}\geq c\sqrt{p}\|P_{\sigma_{x}([2,n])}x\|$, i.e., vectors with relatively big two largest components. For such vectors we produce needed anti-concentration estimates for the matrix-vector products by using only those two components, i.e., we consider anti-concentration for the vector $P_{A}x$, where $A=\sigma_{x}(\{1,2\})$. Since the Rogozin lemma is not suitable for this case, we compute the anti-concentration directly in Proposition 3.11. As for the classes $\mathcal{B}_{2},\mathcal{B}_{3}$, we actually intersect the fourth class with appropriately chosen sets of almost constant vectors in order to control cardinalities of the nets. The final step is to show that the set ${\mathcal{S}}_{n}$ is contained in the union of four sets described here. Careful analysis of this approach shows that the result can be proved with all constants and parameters $r,\delta,\rho$ depending only on $q$. Thus, it works for $p$ being between the two constants $q$ and $c$.

The case of small $p$, that is, the case $C(\ln n)/n\leq p\leq c$, requires a more sophisticated splitting of ${\mathcal{S}}_{n}$ — we split it into steep vectors and ${\mathcal{R}}$-vectors. The definition and the treatment of steep vectors essentially follows [24, 27], with corresponding adjustments for our model. The set of steep vectors consists of vectors having a large jump between order statistics measured at certain indices. The first subclass of steep vectors, $\mathcal{T}_{0}$, is the same as the class $\mathcal{B}_{1}$ described above — vectors having very large maximal coordinate — and is treated as $\mathcal{B}_{1}$. Similarly to the case of constant $p$, this class is the main contributor to the bound $(1+o_{n}(1))n(1-p)^{n}$ for non-invertibility over structured vectors. Next we fix certain $m\approx 1/p$ and consider a sequence $n_{0}=2$, $n_{j+1}/n_{j}=\ell_{0}$, $j\leq s_{0}-1$, $n_{s_{0}+1}=m$ for some specially chosen parameters $\ell_{0}$ and $s_{0}$ depending on $p$ and $n$. The class $\mathcal{T}_{1}$ will be defined as the class of vectors such that there exists $j$ with $x^{*}_{n_{j+1}}>6pnx^{*}_{n_{j}}$. To work with vectors from this class, we first show that for a given $j$ the event that for every choice of two disjoint sets $|J_{1}|=n_{j}$ and $|J_{2}|=n_{j+1}-n_{j}$, a random Bernoulli($p$) matrix has a row with exactly one $1$ in components indexed by $J_{1}$ and no $1$’s among components indexed by $J_{2}$, holds with a very high probability. Then, conditioned on this event, for every $x\in\mathcal{T}_{1}$, we choose $J_{1}$ corresponding to $x_{i}^{*}$, $i\leq n_{j}$, and $J_{2}$ corresponding to $x_{i}^{*}$, $n_{j}\leq i\leq n_{j+1}$, and the corresponding row. Then the inner product of this row with $x$ will be large in absolute value due to the jump (see Lemma 6.9 for the details). Thus, conditioned on the described event, for every $x\in\mathcal{T}_{1}$ we have a good lower bound on $\|Mx\|$. Then next two classes of steep vectors, $\mathcal{T}_{2}$ and $\mathcal{T}_{3}$, consist of vectors having a jump of order $C\sqrt{pn}$, namely, vectors in $\mathcal{T}_{2}$ satisfy $x_{m}^{*}>C\sqrt{pn}x_{k}^{*}$ and vectors in $\mathcal{T}_{3}$ satisfy $x_{k}^{*}>C\sqrt{pn}x_{\ell}^{*}$, where $k\approx\sqrt{n/p}$ and $\ell=\lfloor rn\rfloor$ ($r$ is the parameter from the definition of ${\mathcal{V}}_{n}(r,{\bf g},\delta,\rho)$). Trying to apply the same idea for these two subclasses one sees that the size of corresponding sets $J_{1}$ and $J_{2}$ is too large to have exactly one $1$ among a row’s components indexed by $J_{1}\cup J_{2}$ with a high probability. Therefore the proof of individual probability bounds is more delicate and technical as well as a construction of corresponding nets for $\mathcal{T}_{2},\mathcal{T}_{3}$. We discuss the details in Subsection 6.6.

The class of ${\mathcal{R}}$-vectors consists of non-steep vectors to which Rogozin’s lemma (Proposition 3.9) can be applied when we project a vector on $n-k$ smallest coordinates with $m<k\leq n/\ln^{2}(pn)$, thus vectors from this class satisfy $\|P_{A}x\|\leq c\sqrt{p}\|P_{A}x\|_{\infty}$ for $A=\sigma_{x}([k,n]$ (we will take union over all choices of integer $k$ in the interval $(m,n/\ln^{2}(pn)]$). Thus, the individual probability bounds for ${\mathcal{R}}$-vectors will follow from Rogozin’s lemma together with tensorization lemma as for classes $\mathcal{B}_{2}$, $\mathcal{B}_{3}$, described above. Thus the remaining part is to construct a good net for ${\mathcal{R}}$-vectors. For simplicity, dealing with such vectors, we fix the normalization $x^{*}_{\lfloor rn\rfloor}=1$. Since vectors are non-steep, we have a certain control of largest coordinates and, thus, on the Euclidean norm of a vector. The upper bound on $k$ is chosen in such a way that the cardinality on a net corresponding to largest coordinates of a vector is relatively small (it lies in $n/\ln^{2}(pn)$-dimensional subspace). For the purpose of constructing of a net of small cardinality, we need to control the Euclidean norm of $P_{A}x$ for an ${\mathcal{R}}$-vector. Therefore we split ${\mathcal{R}}$-vectors into level sets according to the value of $\|P_{A}x\|$. There will be two different types of level sets — vectors with relatively large Euclidean norm of $P_{A}x$ and vectors with small $\|P_{A}x\|$. A net for level sets with large $\|P_{A}x\|$ is easier to construct, since we can zero all coordinates starting with $x^{*}_{\lfloor rn\rfloor}=1$. If the Euclidean norm is small, we cannot do this, so we intersect this subclass with almost constant vectors (in fact we incorporate this intersection into the definition of ${\mathcal{R}}$-vectors), defined by

$$\mathcal{AC}(\rho):=\{x\in{\mathbb{R}}^{n}\,:\,\exists\lambda\in{\mathbb{R}}\,\mbox{ s. t. }\,|\lambda|=x^{*}_{\lfloor rn\rfloor}\,\mbox{ and }\,|\{i\leq n\,:\,|x_{i}-\lambda|\leq\rho|\lambda|\}|>n-\lfloor rn\rfloor\}.$$

(9)

As in the case of constant $p$, this essentially reduces the dimension corresponding to almost constant part to one and therefore reduce the cardinality of a net. The rather technical construction of nets is presented in Subsection 6.3. In some aspects the construction follows ideas developed in [24].

By universal or absolute constants we always mean numbers independent of all involved parameters, in particular independent of $p$ and $n$. Given positive integers $\ell<k$ we denote sets $\{1,2,\ldots,\ell\}$ and $\{\ell,\ell+1,\ldots,k\}$ by $[\ell]$ and $[\ell,k]$ correspondingly. Having two functions $f$ and $g$ we write $f\approx g$ if there are two absolute positive constants $c$ and $C$ such that $cf\leq g\leq Cf$. As usual, $\Pi_{n}$ denotes the permutation group on $[n]$.

For every vector $x=(x_{i})_{i=1}^{n}\in{\mathbb{R}}^{n}$, by $(x_{i}^{*})_{i=1}^{n}$ we denote the non-increasing rearrangement of the sequence $(|x_{i}|)_{i=1}^{n}$ and we fix one permutation $\sigma_{x}$ satisfying $|x_{\sigma_{x}(i)}|=x_{i}^{*}$, $i\leq n$. We use $\left\langle\cdot,\cdot\right\rangle$ for the standard inner product on ${\mathbb{R}}^{n}$, that is $\left\langle x,y\right\rangle=\sum_{i=1}^{n}x_{i}y_{i}$. Further, we write $\|x\|_{\infty}=\max_{i}|x_{i}|$ for the $\ell_{\infty}$-norm of $x$. We also denote ${\bf 1}=(1,1,\dots,1)$.

Here, we provide a simple argument showing that for the sequence of random Bernoulli($p_{n}$) matrices $(M_{n})$, with $p_{n}$ satisfying $(np_{n}-\ln n)\longrightarrow\infty$ as $n\to\infty$, we have

$${\mathbb{P}}\big{\{}\mbox{$M_{n}$ contains a zero row or column}\big{\}}\geq(2-o_{n}(1))n\,(1-p)^{n}.$$

Our approach is similar to that applied in [5] in the related context.

Fix $n>1$ and write $p=p_{n}$. Let ${\bf 1}_{R}$ be the indicator of the event that there is zero row in the matrix $M_{n}$, and, similarly, let ${\bf 1}_{C}$ be the indicator of the event that $M_{n}$ has a zero column. Then, obviously,

$${\mathbb{E}}\,{\bf 1}_{R}={\mathbb{E}}\,{\bf 1}_{C}=1-\big{(}1-(1-p)^{n}\big{)}^{n},$$

hence,

$${\mathbb{E}}({\bf 1}_{R}+{\bf 1}_{C})^{2}\geq 2-2\big{(}1-(1-p)^{n}\big{)}^{n}.$$

On the other hand,

$${\mathbb{E}}\,{\bf 1}_{R}\,{\bf 1}_{C}\leq\sum_{i=1}^{n}\sum_{j=1}^{n}{\mathbb{P}}\big{\{}\mbox{$i$--th row and $j$--th column of $M_{n}$ are zero}\big{\}}=n^{2}(1-p)^{2n-1},$$

implying

$${\mathbb{E}}({\bf 1}_{R}+{\bf 1}_{C})^{2}={\mathbb{P}}\big{\{}{\bf 1}_{R}+{\bf 1}_{C}=1\big{\}}+4\,{\mathbb{P}}\big{\{}{\bf 1}_{R}\,{\bf 1}_{C}=1\big{\}}\leq{\mathbb{P}}\big{\{}{\bf 1}_{R}+{\bf 1}_{C}=1\big{\}}+4n^{2}(1-p)^{2n-1}.$$

Therefore,

	$\displaystyle{\mathbb{P}}\big{\{}\mbox{$M_{n}$ contains a zero row or column}\big{\}}$	$\displaystyle\geq{\mathbb{P}}\big{\{}{\bf 1}_{R}+{\bf 1}_{C}=1\big{\}}$
		$\displaystyle\geq{\mathbb{E}}({\bf 1}_{R}+{\bf 1}_{C})^{2}-4n^{2}(1-p)^{2n-1}$
		$\displaystyle\geq 2-2\big{(}1-(1-p)^{n}\big{)}^{n}-4n^{2}(1-p)^{2n-1}.$

It remains to note that, with our assumption on the growth rate of $p=p_{n}$, we have $n(1-p)^{n}\longrightarrow 0$, which implies

$$\frac{1}{n(1-p)^{n}}\big{(}2-2\big{(}1-(1-p)^{n}\big{)}^{n}-4n^{2}(1-p)^{2n-1}\big{)}\longrightarrow 2.$$

For any $r\in(0,1)$, we define ${\Upsilon}_{n}(r)$ as the set of all vectors $x$ in ${\mathbb{R}}^{n}$ with $x^{*}_{\lfloor rn\rfloor}=1$. We will call these vectors $r$-normalized. By a growth function ${\bf g}$ we mean any non-decreasing function from $[1,\infty)$ into $[1,\infty)$.

Let ${\bf g}$ be an arbitrary growth function. We will say that a vector $x\in{\Upsilon}_{n}(r)$ is gradual (with respect to the function ${\bf g}$) if $x^{*}_{i}\leq{\bf g}(n/i)$ for all $i\leq n$. Further, if $x\in{\Upsilon}_{n}(r)$ satisfies

$\displaystyle\exists\,Q_{1},Q_{2}\subset[n]\quad\mbox{ such that }\quad|Q_{1}|,|Q_{2}|\geq\delta n\quad\mbox{ and }\quad\max\limits_{i\in Q_{2}}x_{i}\leq\min\limits_{i\in Q_{1}}x_{i}-\rho$

(10)

then we say that the vector $x$ is essentially non-constant or just non-constant (with parameters $\delta,\rho$). Recall that the set ${\mathcal{V}}_{n}={\mathcal{V}}_{n}(r,{\bf g},\delta,\rho)$ was defined in (2) as

$$\big{\{}x\in{\Upsilon}_{n}(r):\,\,x\,\mbox{ is gradual with ${\bf g}$ and satisfies \eqref{cond2}}\big{\}}.$$

Vectors from this set we call gradual non-constant vectors.