On unique recovery of finite-valued integer signals and admissible lattices of sparse hypercubes

Abstract

The paper considers the problem of unique recovery of sparse finite-valued integer signals using a single linear integer measurement. For l-sparse signals in ${\mathbb {Z}}^n$, $2l<n$, with absolute entries bounded by r, we construct an $1\times n$ measurement matrix with maximum absolute entry $\Delta =O(r^{2l-1})$. Here the implicit constant depends on l and n and the exponent $2l-1$ is optimal. Additionally, we show that, in the above setting, a single measurement can be replaced by several measurements with absolute entries sub-linear in $\Delta$. The proofs make use of results on admissible $(n-1)$-dimensional integer lattices for m-sparse n-cubes that are of independent interest.

1 Unique recovery of sparse bounded integer signals

Fix a set $S\subset {\mathbb {Z}}^n$ which will be referred to as a signal space. We will consider the problem of unique recovery of a signal ${\varvec{x}}_0\in S$ from a relatively small number of noisy linear integer measurements, in the form introduced by Fukshansky et al. [9]. Specifically, given a number of measurements m with $m<n$, we aim to construct an integer measurement matrix $A\in {\mathbb {Z}}^{m\times n}$ such that any signal ${\varvec{x}}_0\in S$ can be uniquely recovered from m measurements represented by the vector ${\varvec{b}}\in {\mathbb {R}}^m$ of the form

$$\begin{aligned} {\varvec{b}} = A{\varvec{x}}_0+{\varvec{e}} \end{aligned}$$

with an unknown noise vector ${\varvec{e}}\in {\mathbb {R}}^m$. To allow unique recovery we assume that

$$\begin{aligned} \Vert {\varvec{e}}\Vert _2< c\,, \end{aligned}$$

where $\Vert \cdot \Vert _{2}$ denotes the $\ell _2$-norm and c is a suitably chosen constant.

Based on [9], we will use the following recovery approach. For a set $Q\subset {\mathbb {Z}}^n$ we denote by ${{\mathcal {R}}}(Q)$ the set of all matrices $A\in {\mathbb {Z}}^{k\times n}$ with $k<n$, such that

$$\begin{aligned} \Vert A{\varvec{y}}\Vert _2 \ge 1 \text{ for } \text{ any } \text{ nonzero } {\varvec{y}}\in Q\,. \end{aligned}$$

(1)

Recall that the difference set D(X) of a set $X\in {\mathbb {R}}^n$ consists of all points ${\varvec{x}}-{\varvec{y}}$ with ${\varvec{x}}, {\varvec{y}}\in X$. We set $c=1/2$ and consider $m\times n$ measurement matrices $A\in {{\mathcal {R}}}(Q)$ with $Q=D(S)$. In this case, for any ${\varvec{e}}\in {\mathbb {R}}^m$ with $\Vert {\varvec{e}}\Vert _2<c=1/2$, the signal ${\varvec{x}}_0$ is the unique point of S satisfying the bound

$$\begin{aligned} \Vert A{\varvec{x}}-{\varvec{b}}\Vert _2 \le \frac{1}{2}\,. \end{aligned}$$

(2)

Indeed, for any ${\varvec{x}}\in S$, ${\varvec{x}}\ne {\varvec{x}}_0$, satisfying (2), we would have

$$\begin{aligned} \begin{aligned} \Vert A({\varvec{x}}-{\varvec{x}}_0)\Vert _2&= \Vert A{\varvec{x}}-{\varvec{b}}-(A{\varvec{x}}_0-{\varvec{b}})\Vert _2\\&= \Vert A{\varvec{x}}-{\varvec{b}}+{\varvec{e}}\Vert _2 \\&\le \Vert A{\varvec{x}}-{\varvec{b}}\Vert _2+\Vert {\varvec{e}}\Vert _2< 1\,, \end{aligned} \end{aligned}$$

contradicting (1). Therefore, ${\varvec{x}}_0$ can be recovered by any algorithm that, given input ${\varvec{b}}\in {\mathbb {R}}^m$ computes a vector ${\varvec{x}}\in S$ satisfying (2).

We will now introduce some basic notation needed for stating our results. Given ${\varvec{x}}=(x_1, \ldots , x_n)^T\in {\mathbb {R}}^n$, we will denote by $\Vert {\varvec{x}}\Vert _0=|\{i : x_i \not = 0\}|$ the 0-“norm”, widely used in the theory of compressed sensing [5, 6], which counts the cardinality of the support of ${\varvec{x}}$. A vector ${\varvec{x}}\in {\mathbb {R}}^n$ is called l-sparse if $\Vert {\varvec{x}}\Vert _0\le l$. By ${\mathbb {Z}}^n_l$ we denote the set of l-sparse n-dimensional integer vectors:

$$\begin{aligned} {\mathbb {Z}}^n_l=\{{\varvec{z}}\in {\mathbb {Z}}^n: \Vert {\varvec{z}}\Vert _0\le l\}\,. \end{aligned}$$

Given positive integers n, r, we denote by $C^n(r)$ the n-dimensional cube defined as $C^n(r)=\{ {\varvec{x}}\in {\mathbb {R}}^n: \Vert {\varvec{x}}\Vert _{\infty }\le r \}$, where $\Vert \cdot \Vert _{\infty }$ stands for the $\ell _\infty$-norm.

We will be interested in unique recovery of l-sparse signals with entries from a finite integer alphabet $[-r,r]\cap {\mathbb {Z}}$. Specifically, we will work with the signal space

$$\begin{aligned} S^n_l(r)=C^n(r) \cap {\mathbb {Z}}^n_l\,, \end{aligned}$$

where $2l< n$.

The signal space $S^n_l(r)$ is finite and hence allows using a single measurement for unique recovery of its signals. From the computational and error-correcting perspectives (see [9] for more details), the measurement should have as small as possible absolute integer entries. Hence, given $l, n\in {\mathbb {Z}}$ with $1\le l <n/2$, $r\in {\mathbb {Z}}_{>0}$ and letting $Q=D(S^n_l(r))$, we face the optimisation problem

$$\begin{aligned} \min \{\Vert H\Vert _\infty : H\in {\mathbb {Z}}^{1\times n}, \; H\in {{\mathcal {R}}}(Q)\}\,. \end{aligned}$$

(3)

In this paper, we will obtain general estimates for the minimum in (3). Using condition (1), to get an upper bound for (3), it is sufficient to find an $1\times n$ measurement matrix H such that its kernel space does not share any nonzero integer points with the convex hull of $D(S^n_l(r))$. This straightforward approach, however, results in the estimate

$$\begin{aligned} \Vert H\Vert _{\infty }=O(r^{n-1})\,, \end{aligned}$$

(4)

where the implicit constant depends on l and n.

The first result of this paper shows that the exponent $n-1$ in (4) can be replaced with $2l-1$. Let

$$\begin{aligned} p_{m,n}(r)=(m\Delta (m,n)r+1)^{m-1} +\Delta (m,n) \sum _{i=0}^{m-2} (m\Delta (m,n)r+1)^{i}\,, \end{aligned}$$

where $\Delta (m,n) = \min (n-1, \lceil (2n)^{(m-1)/m}\rceil )$.

Theorem 1

For any $l, n\in {\mathbb {Z}}$ with $1\le l <n/2$ and $r\in {\mathbb {Z}}_{>0}$ there exists an $1\times n$ integer matrix H such that $H\in {{\mathcal {R}}}(Q)$ with $Q=D(S^n_l(r))$ and

$$\begin{aligned} \Vert H\Vert _{\infty }\le p_{2l,n}(2r) \,. \end{aligned}$$

(5)

The proof of Theorem 1 is constructive. To obtain the bound (5) we combine known results on unique recovery over ${\mathbb {Z}}^n_l$, outlined in Sect. 2, with aggregation techniques, outlined in Sect. 4.

The second result gives a lower bound for the minimum in (3). Notably, it shows that the polynomial $p_{2l,n}(2r)$ in (5) cannot be replaced by a polynomial in r with degree smaller than ${2l-1}$.

Theorem 2

For any $l, n\in {\mathbb {Z}}$ with $1\le l <n/2$, $r\in {\mathbb {Z}}_{>0}$ and $1\times n$ integer matrix $H\in {{\mathcal {R}}}(Q)$ with $Q=D(S^n_l(r))$ the bound

$$\begin{aligned} \Vert H\Vert _{\infty }> \frac{r^{2l-1}}{\sqrt{2l}} \end{aligned}$$

(6)

holds.

Based on Theorems 1 and 2 we pose the following question. Let us fix the sparsity level l and dimension n. In this setting, it would be interesting to find optimal upper bounds for minimal $\Vert H\Vert _{\infty }/r^{2l-1}$ when r tends to infinity. Specifically, given $l, n \in {\mathbb {Z}}_{>0}$ with $1\le l <n/2$, to estimate

$$\begin{aligned} c_1(l,n)=\limsup \inf \frac{\Vert H\Vert _{\infty }}{r^{2l-1}}\,, \end{aligned}$$

(7)

where the supremum limit is taken over all positive integers r and the infimum is taken over all $1\times n$ integer matrices $H\in {{\mathcal {R}}}(Q)$ with $Q=D(S^n_l(r))$. Theorems 1 and 2 give a large interval for values of this quantity

$$\begin{aligned} \frac{1}{\sqrt{2l}}\le c_1(l,n)\le (4l\Delta (2l,n))^{2l-1}\,. \end{aligned}$$

Although for finite signal spaces a single measurement is sufficient for unique recovery, one can ask whether allowing extra measurements would result in reducing measurements’ entries. In this vein, we obtain the following general result. Let $Q\subset {\mathbb {Z}}^n$ be an arbitrary set. Suppose that we have an $1\times n$ matrix $H\in {{\mathcal {R}}}(Q)$. We show that, for any integer m with $1<m<n$, there exists an $m\times n$ matrix $A=(a_{ij})$ such that $A\in {{\mathcal {R}}}(Q)$ and the maximum absolute entry $\Vert A\Vert _{\infty }=\max _{i,j} |a_{ij}|$ is sub-linear in $\Vert H\Vert _\infty$.

Let $\gamma _{r,s}$ be the generalised Hermite constant, as defined by Rankin [14], that is the least number such that every lattice $\Lambda$ of rank r in ${\mathbb {R}}^r$ has a sublattice $\Gamma$ of rank s and determinant

$$\begin{aligned} \det (\Gamma )\le \gamma _{r,s}^{1/2} (\det (\Lambda ))^{s/r}\,. \end{aligned}$$

Here $\gamma _{r,1}=\gamma _{r,r-1}=\gamma _r$ is the ordinary Hermite constant. For known results on the Rankin constant we refer the reader to the papers [14, 17, 19].

Theorem 3

Let $Q\subset {\mathbb {Z}}^n$ and let H be an $1\times n$ matrix such that $H\in {{\mathcal {R}}}(Q)$. For any integer m with $1<m<n$, there exists an $m\times n$ matrix A such that $A\in {{\mathcal {R}}}(Q)$ and

$$\begin{aligned} \Vert A\Vert _{\infty }\le c_2(m,n) \Vert H\Vert _\infty ^{\frac{n-m}{n-1}}\,, \end{aligned}$$

(8)

where $c_2(m,n)= \gamma _{n-1,n-m}^{1/2} n^{(n-m)/(2(n-1))}$.

The proof of Theorem 3 makes use of results on rational subspaces obtained in [2]. Note that (8) improves the immediate bound $\Vert A\Vert _{\infty }\le \Vert H\Vert _\infty$ when $\Vert H\Vert _\infty > c_2(m,n)^{(n-1)/(m-1)}$.

2 Unique recovery over ${\mathbb {Z}}^n_l$

The papers [8, 9, 11, 12] consider the problem of unique recovery over the signal space ${\mathbb {Z}}^n_l$, where l is a positive integer with $2l< n$. In this setting, the difference set $D({\mathbb {Z}}^n_l)$ consists of 2l-sparse integer vectors, $D({\mathbb {Z}}^n_l)={\mathbb {Z}}^n_{2l}\,.$ The unique recovery of signals from ${\mathbb {Z}}^n_l$ involves constructing matrices $A\in {\mathbb {Z}}^{m\times n}$ with $m=2l$ and as large as possible n, that belong to ${{\mathcal {R}}}({\mathbb {Z}}^n_m)$. From the computational and error-correcting perspectives (see [9] for more details), it is also desirable to fix or bound the maximum absolute entry $\Vert A\Vert _{\infty }$ of the matrix A.

Konyagin [12, Theorem 3] proved the following theorem.

Theorem 4

For integers $k, m\ge 2$, and integer n with

$$\begin{aligned} m<n\le \frac{c^{-m}k^{m/(m-1)}}{\log (k)} \end{aligned}$$

there exists an integer $m\times n$ matrix $A\in {{\mathcal {R}}}({\mathbb {Z}}^n_m)$ such that $\Vert A\Vert _{\infty }=k$, where c is an absolute constant.

The proof of Theorem 4 employs probabilistic arguments to show existence of the desired measurement matrices. Subsequently, Konyagin and Sudakov [11, Theorem 1.3] (see also Ryutin [16]) proved the following result using an explicit and easily computable construction.

Theorem 5

Let $k\in {\mathbb {Z}}_{>0}$, $m\in {\mathbb {Z}}_{>0}$, $m\ge 2$, and

$$\begin{aligned} m<n\le \max (k+1, k^{m/(m-1)}/2)\,. \end{aligned}$$

(9)

Then there is an $m\times n$ integer matrix $A\in {{\mathcal {R}}}({\mathbb {Z}}^n_m)$ such that $\Vert A\Vert _{\infty }\le k$.

Theorem 5 implies the following corollary.

Corollary 6

For any given $m,n\in {\mathbb {Z}}_{>0}$, $2\le m<n$ there exists an $m\times n$ integer matrix $A\in {{\mathcal {R}}}({\mathbb {Z}}^n_m)$ with

$$\begin{aligned} \Vert A\Vert _{\infty } \le \Delta (m,n)=\min (n-1, \lceil (2n)^{(m-1)/m}\rceil )\,. \end{aligned}$$

3 Admissible lattices of m-sparse n-cubes

By a rational subspace of ${\mathbb {R}}^n$ we understand a subspace generated by integer vectors. A rational hyperplane can be written as $P=\{{\varvec{x}}\in {\mathbb {R}}^n: H{\varvec{x}}=0\}$, where $H=(H_{11},\ldots , H_{1n})$ is an $1\times n$ integer matrix with $\gcd (H):=\gcd (H_{11},\ldots , H_{1n})=1$. We say that P has height $h(P)=\Vert H\Vert _\infty$.

For linearly independent ${\varvec{b}}_1, \ldots , {\varvec{b}}_l$ in ${\mathbb {R}}^d$, the set $\Lambda =\{\sum _{i=1}^{l} x_i {\varvec{b}}_i,\, x_i\in {\mathbb {Z}}\}$ is an l-dimensional lattice with basis ${\varvec{b}}_1, \ldots , {\varvec{b}}_l$. Denoting by B the matrix with columns ${\varvec{b}}_1, \ldots , {\varvec{b}}_l$, the determinant of $\Lambda$ is defined as $\det (\Lambda )=\sqrt{\det (B^TB)}$. A lattice $\Lambda \subset {\mathbb {R}}^d$ is (strictly) admissible for a set $X\subset {\mathbb {R}}^d$ if $\Lambda$ does not contain any nonzero point of X, that is $\Lambda \cap X\subset \{ {\varvec{0}} \}$. For a comprehensive introduction to the theory of lattices we refer the reader to [7, 10].

Let r be a positive integer and m be a positive integer with $1<m< n$. We will consider an m-sparse n-dimensional cube

$$\begin{aligned} C^n_m(r)=\{ {\varvec{x}}\in C^n(r): \Vert {\varvec{x}}\Vert _0\le m \}\,. \end{aligned}$$

Constructing single measurements for unique recovery of sparse integer signals is closely linked to constructing admissible $(n-1)$-dimensional lattices for $C^n_m(r)$. From the unique recovery perspective, it is desirable to find a rational hyperplane P of smallest possible height such that the lattice $P\cap {\mathbb {Z}}^n$ is admissible for $C^n_m(r)$. Similarly to (3), we consider the following optimisation problem. Given $m,n\in {\mathbb {Z}}$ with $1<m<n$ and $r\in {\mathbb {Z}}_{>0}$, find

$$\begin{aligned} \begin{aligned} \min \{h(P): P&\text{ is } \text{ a } \text{ rational } \text{ hyperplane } \text{ in } {\mathbb {R}}^n \text{ such } \text{ that } \text{ the } \text{ lattice } P\cap {\mathbb {Z}}^n \text{ is } \\&\text{ admissible } \text{ for } C^n_m(r))\}\,. \end{aligned} \end{aligned}$$

(10)

The proofs of Theorems 1 and 2 will be based on the following estimates for the minimum in (10) that are of independent interest.

Proposition 7

For any $m, n\in {\mathbb {Z}}$ with $1<m <n$ and $r\in {\mathbb {Z}}_{>0}$ there exists a rational hyperplane P in ${\mathbb {R}}^n$ such that the lattice $P\cap {\mathbb {Z}}^n$ is admissible for $C^n_m(r)$ and

$$\begin{aligned} h(P)\le p_{m,n}(r) \,. \end{aligned}$$

(11)

To prove Proposition 7 we combine constructions from the proof of Theorem 5 with aggregation techniques outlined in Sect. 4. The next result shows that the polynomial $p_{m,n}(r)$ in (11) cannot be replaced by a polynomial in r of degree smaller than ${m-1}$.

Proposition 8

For any $m, n\in {\mathbb {Z}}$ with $1<m <n$, any $r\in {\mathbb {Z}}_{>0}$ and any rational hyperplane P in ${\mathbb {R}}^n$ such that the lattice $P\cap {\mathbb {Z}}^n$ is admissible for $C^n_m(r)$ the bound 

$$\begin{aligned} h(P)> \frac{r^{m-1}}{\sqrt{m}} \end{aligned}$$

(12)

holds.

Similarly to (7), for $m, n\in {\mathbb {Z}}$ with $1<m <n$, it would be interesting to estimate

$$\begin{aligned} c_3(m,n)=\limsup \inf \frac{h(P)}{r^{m-1}}\,, \end{aligned}$$

where the supremum limit is taken over all positive integers r and the infimum is taken over rational hyperplanes P in ${\mathbb {R}}^n$ such that $P\cap {\mathbb {Z}}^n$ is admissible for $C^n_m(r)$. Propositions 7 and 8 imply the bounds

$$\begin{aligned} \frac{1}{\sqrt{m}}\le c_3(m,n)\le (m\Delta (m,n))^{m-1}\,. \end{aligned}$$

4 Consolidation/aggregation of linear Diophantine equations

The proof of Proposition 7 is based on consolidation/aggregation of linear Diophantine equations. This topic has been extensively studied in the literature. We refer the reader to the papers [13, 15] and references within.

Let $D\subset {\mathbb {Z}}^n$ be a set of integer points, $A\in {\mathbb {Z}}^{m\times n}$, $2\le m<n$, be a matrix of rank $\mathrm {rank}(A)=m$, and ${\varvec{b}}\in {\mathbb {Z}}^m$.

Let $B\in {\mathbb {Z}}^{l\times m}$, $l<m$, be a matrix of rank l such that

$$\begin{aligned} \{{\varvec{x}}\in D: (BA){\varvec{x}}-B{\varvec{b}}={\varvec{0}}\}=\{{\varvec{x}}\in D: A{\varvec{x}}-{\varvec{b}}={\varvec{0}}\}\,. \end{aligned}$$

Following [15], we will call B an m-into-l consolidating matrix and $(BA){\varvec{x}}-B{\varvec{b}}={\varvec{0}}$ an m-into-l consolidation of $A{\varvec{x}}-{\varvec{b}}={\varvec{0}}$ with respect to the set D.

Let further $C\in {\mathbb {Z}}^{m\times (m-l)}$ be an integer matrix of $\mathrm {rank}(C)=m-l$ such that, for some consolidating matrix B, the columns of C span the kernel $\ker (B)=\{ {\varvec{x}}\in {\mathbb {R}}^m: B{\varvec{x}}={\varvec{0}}\}$. That is, denoting by $\mathrm {span}_{{\mathbb {R}}}(C)$ the subspace spanned by the columns of C, we have $\mathrm {span}_{{\mathbb {R}}}(C)=\ker (B)$. We will call C an aggregating matrix for $A{\varvec{x}}-{\varvec{b}}={\varvec{0}}$ with respect to the set D.

We will write

$$\begin{aligned} F({\varvec{x}}) = A{\varvec{x}}-{\varvec{b}}\,, \end{aligned}$$

and denote by $F_i({\varvec{x}})$ the ith entry of the vector $F({\varvec{x}})$, that is $F({\varvec{x}}) =(F_1({\varvec{x}}), \ldots , F_m({\varvec{x}}))^\top.$

Consider the set

$$\begin{aligned} F^o =\{F({\varvec{x}}) : {\varvec{x}}\in D\}= \{A{\varvec{x}}: {\varvec{x}}\in D\}-{\varvec{b}}\,. \end{aligned}$$

This is the image of D under the linear mapping determined by the matrix A translated by the vector $-{\varvec{b}}$. The following well-known lemma describes very important properties of the consolidation/aggregation.

Lemma 9

Let $B\in {\mathbb {Z}}^{l\times m}$, $l<m$, be a matrix of rank l and $C\in {\mathbb {Z}}^{m\times (m-l)}$ be a matrix of rank $m-l$. Then

• (i) B an m-into-l consolidating matrix for $A{\varvec{x}}-{\varvec{b}}={\varvec{0}}$ if and only if $F^o\cap \ker (B)\subset \{\varvec{0}\}$.

• (ii) C is an aggregating matrix of $A{\varvec{x}}-{\varvec{b}}={\varvec{0}}$ if and only if $F^o\cap \mathrm {span}_{{\mathbb {R}}}(C)\subset \{\varvec{0}\}$.

We will need the following lemma, given in [3, Theorem 6]. For completeness, we include a proof of this result as given in [15, Example 4.1].

Lemma 10

Assume that $q_i\in {\mathbb {Z}}$ satisfy $|F_i({\varvec{x}})|<q_i$ for every ${\varvec{x}}\in D$ such that $F_1({\varvec{x}})=\cdots =F_{i-1}({\varvec{x}})=0$, $i=1,\ldots , m-1$. Then $F_1({\varvec{x}})+q_1F_2({\varvec{x}})+q_1q_2F_3({\varvec{x}})+\cdots +q_1\cdots q_{m-1}F_m({\varvec{x}})={0}$ is an m-into-1 consolidation of $F({\varvec{x}})={\varvec{0}}$ with respect to the set D.

Proof

We have to show that $B=(1, q_1, \ldots , q_1\cdots q_{m-1})$ is an m-into-1 consolidating matrix for $F({\varvec{x}})={\varvec{0}}$. Let $C=(c_{ij})\in {\mathbb {Z}}^{m\times (m-1)}$ be defined by $c_{1,k}=q_1 \delta _{1,k}$ for $k=1,\ldots , m-1$ and $c_{ij}=q_i\delta _{i,j}-\delta _{i-1,j}$ for $i=2,\ldots , m$, $j=1,\ldots , m-1$. Here $\delta _{i,j}$ stands for the Kronecker delta. That is

$$\begin{aligned} C=\left( \begin{array}{rrrrr} q_1 &{} 0 &{} \cdots &{} 0 &{} 0\\ -1 &{} q_2 &{} \cdots &{} 0 &{} 0\\ 0 &{} -1 &{} \cdots &{} 0&{} 0\\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} -1 &{} q_{m-1}\\ 0 &{} 0 &{} \cdots &{} 0 &{} -1 \end{array} \right) \,. \end{aligned}$$

We will show that C is an aggregating matrix for $F({\varvec{x}})={\varvec{0}}$. It is sufficient to check that the inclusion $F^o\cap \mathrm {span}_{{\mathbb {R}}}(C)\subset \{\varvec{0}\}$ in the part (ii) of Lemma 9 holds. Suppose

$$\begin{aligned} C {\varvec{v}}=F({\varvec{x}}) \end{aligned}$$

(13)

for some ${\varvec{v}}\in {\mathbb {R}}^{m-1}$ and ${\varvec{x}}\in D$.

Observe that the greatest common divisor of $(m-1)\times (m-1)$ subdeterminants of C is equal to one. It follows that the columns of C form a basis of the lattice $\mathrm {span}_{{\mathbb {R}}}(C)\cap {\mathbb {Z}}^m$. Next, by (13) we have $C {\varvec{v}}\in {\mathbb {Z}}^m$. Therefore, ${\varvec{v}}\in {\mathbb {Z}}^{m-1}$. The first coordinate of $C {\varvec{v}}$ is $q_1v_1$, hence $|q_1v_1|=|F_1({\varvec{x}})|<|q_1|$ implies $v_1=F_1({\varvec{x}})=0$. The second coordinate of $C {\varvec{v}}$ is $q_2v_2-v_1=q_2v_2=F_2({\varvec{x}})$. Now by definition of $q_2$ we have $|q_2v_2|=|F_2({\varvec{x}})|<|q_2|$ and, consequently, $v_2=0$. Proceeding in this way we get ${\varvec{v}}={\varvec{0}}$.

Finally, it is easy to see that the columns of C span $\ker (B)$. Hence B is an m-into-1 consolidating matrix for $F({\varvec{x}})={\varvec{0}}$. $\square$

5 Proofs of Proposition 7 and Theorem 1

We will begin with proving Proposition 7. The proof of Theorem 5 (Konyagin and Sudakov [11, Theorem 1.3]) gives two explicit constructions that can be used to obtain matrices $A\in {{\mathcal {R}}}({\mathbb {Z}}^n_m)$ that satisfy conditions of Corollary 6. For completeness, we will outline these constructions here.

Observe first that $A\in {{\mathcal {R}}}({\mathbb {Z}}^n_m)$ if and only if all $m\times m$ subdeterminants of A are nonzero. Therefore, if for some d satisfying $m<n<d$ there exists an $m\times d$ matrix $A\in {{\mathcal {R}}}({\mathbb {Z}}^d_m)$, then an $m\times n$ matrix in ${{\mathcal {R}}}({\mathbb {Z}}^n_m)$ can be obtained by removing any $d-n$ columns from A. Set first $k=n-1$. The first construction gives $A=(a_{ij})\in {{\mathcal {R}}}({\mathbb {Z}}^d_m)$ with $d\ge k+1$. The dimension d is chosen as an odd prime number satisfying $k+1\le d \le 2k+1$. Subsequently, the entries of the matrix A are defined as $a_{ij}\equiv j^{i-1} (\bmod \; d)$ with $|a_{ij}|\le (d-1)/2\le k$. In particular, for all j we have $a_{1j}=1$. Next, set $k=\lceil (2n)^{(m-1)/m} \rceil$. The second construction gives $A=(a_{ij})\in {{\mathcal {R}}}({\mathbb {Z}}^d_m)$ with $d\ge k^{m/(m-1)}/2$. The dimension d is chosen as a prime number with $k^{m/(m-1)}/2\le d \le k^{m/(m-1)}$. The entries of the matrix A satisfy $a_{ij}\equiv l_{ij} j^{i-1} (\;\bmod \; d)$, where $l_{ij}$ are certain integers not divisible by d chosen in a such way that $|a_{ij}|\le k$. In particular, for all j one can take $l_{1j}=1$, so that $a_{1j}=1$.

In both constructions above, renumbering the rows of A, we may assume that $a_{mj}=1$ for all j. Set $k=\Delta (m,n)$ and for $s= m\, k\, r +1\,$ take

$$\begin{aligned} B= (1, s, \ldots , s^{m-1})\, \end{aligned}$$

and

$$\begin{aligned} H= BA\,. \end{aligned}$$

We will show that the hyperplane $P=\ker (H)$ satisfies the conditions of Proposition 7.

Let $F({\varvec{x}})= A{\varvec{x}}$ and let $F_i({\varvec{x}})$ denote the ith entry of $F({\varvec{x}})$, that is

$$\begin{aligned} \begin{array}{l} F_1({\varvec{x}}) = a_{11}x_1+\cdots +a_{1n}x_n\,,\\ \;\;\;\vdots \\ F_m({\varvec{x}}) = a_{m1}x_1+\cdots +a_{mn}x_n\,. \end{array} \end{aligned}$$

For any ${\varvec{x}}\in C^n_m(r)$ and any $i\in \{1,\ldots , m\}$, we have

$$\begin{aligned} F_i({\varvec{x}}) \le \Vert {\varvec{x}}\Vert _0\Vert A\Vert _{\infty }r\le m\,k\,r< s\,. \end{aligned}$$

Lemma 10, applied with $D= C^n_m(r)\cap {\mathbb {Z}}^n$ and $q_i=s$ for $i=1,\ldots , m-1$, implies that

$$\begin{aligned} \{{\varvec{x}}\in C^n_m(r)\cap {\mathbb {Z}}^n: H{\varvec{x}}= {\varvec{0}}\} =\{{\varvec{x}}\in C^n_m(r)\cap {\mathbb {Z}}^n: A{\varvec{x}}= {\varvec{0}}\}\,. \end{aligned}$$

(14)

Since $A\in {{\mathcal {R}}}( {\mathbb {Z}}^n_{m})$ we have

$$\begin{aligned} \{{\varvec{x}}\in C^n_m(r)\cap {\mathbb {Z}}^n: A{\varvec{x}}= {\varvec{0}}\}= \{{\varvec{0}}\}\,. \end{aligned}$$

(15)

Consequently, combining (14) and (15), the lattice $P\cap {\mathbb {Z}}^n$ is admissible for $C^n_m(r)$.

Finally, we obtain the bound

$$\begin{aligned}\begin{aligned} h(P)\le \Vert H\Vert _{\infty }\le s^{m-1} + k \sum _{i=0}^{m-2}s^i \end{aligned} \end{aligned}$$

that implies (11).

Remark 1

For given sparsity level m, dimension n and cube size r the set $\{F({\varvec{x}}): {\varvec{x}}\in D\}$ constructed in the proof will likely allow a more accurate choice of parameters $q_i$ in Lemma 10, resulting in an improvement on the bound (11). Further, aggregation techniques can be also applied to the matrices in ${{\mathcal {R}}}({\mathbb {Z}}^n_m)$ obtained using a probabilistic approach from the proof of Theorem 4 (Konyagin [12, Theorem 3]).

5.1 Proof of Theorem 1

Let $m=2l$. By Proposition 7, there is a rational hyperplane P in ${\mathbb {R}}^n$ such that the lattice $P\cap {\mathbb {Z}}^n$ is admissible for $C^n_m(2r)\cap {\mathbb {Z}}^n$ and the bound

$$\begin{aligned} h(P)\le p_{m,n}(2r) \end{aligned}$$

(16)

holds.

We can write $P=\ker (H)$ for an $1\times n$ integer matrix H with $h(P)=\Vert H\Vert _\infty$. The inclusion

$$\begin{aligned} D(S^n_l(r))\subset S^n_m(2r)=C^n_m(2r)\cap {\mathbb {Z}}^n \end{aligned}$$

implies the condition (1) with $A=H$ and $Q=D(S^n_l(r))$. Hence $H\in {{\mathcal {R}}}(Q)$. Finally, the bound (5) immediately follows from (16).

6 Proofs of Proposition 8 and Theorem 2

We will first prove Proposition 8. Let $A \in {\mathbb {Z}}^{m\times n}$, $m<n$, and let $\tau =\{i_1,\ldots ,i_k\} \subseteq \{1,\ldots ,n\}$ with $i_1<i_2<\cdots <i_k$. We will denote by $A_\tau$ the $m \times k$ submatrix of A with columns indexed by $\tau$. In the same manner, given ${\varvec{x}}\in {\mathbb {R}}^n$, we will denote by ${\varvec{x}}_{\tau }$ the vector $(x_{i_1}, \ldots , x_{i_k})^{\top }$. The complement of $\tau$ in $\{1,\ldots , n\}$ will be denoted as ${{\bar{\tau }}}$. For matrices A of rank m, the notation $\gcd (A)$ will be used for the greatest common divisor of all $m \times m$ subdeterminants of A.

The proof makes use of the following version of Siegel’s Lemma obtained by Bombieri and Vaaler [4, Theorem 2].

Theorem 11

Let $M\in {\mathbb {Z}}^{m\times n}$, $m<n$, be a matrix of rank m. There exist $n - m$ linearly independent integer vectors ${\varvec{y}}_1, \ldots ,{\varvec{y}}_{n-m} \in \ker (M)$ satisfying

$$\begin{aligned} \prod _{i=1}^{n-m} \Vert {\varvec{y}}_i\Vert _{\infty } \le \frac{\sqrt{\mathrm{det}(MM^{T})}}{\gcd (M)}\,. \end{aligned}$$

Suppose, to derive a contradiction, that Proposition 8 does not hold. Then for some $m, n\in {\mathbb {Z}}_{>0}$ with $1<m <n$, and $r\in {\mathbb {Z}}_{>0}$ there exists a rational hyperplane P in ${\mathbb {R}}^n$ such that $P\cap {\mathbb {Z}}^n$ is admissible for $C^n_m(r)$ and

$$\begin{aligned} h(P)\le \frac{r^{m-1}}{\sqrt{m}}\,. \end{aligned}$$

(17)

There exists an $1\times n$ integer matrix H such that $P=\ker (H)$ and $h(P)=\Vert H\Vert _\infty$. Take $\tau = \{1, \ldots , m\}$. Observe that H cannot have zero entries, as otherwise its kernel P would contain the corresponding standard basis vectors. Hence, $H_{\tau }\ne {\varvec{0}}$. By Theorem 11, applied with $M=H_{\tau }$, there exists an integer vector ${\varvec{x}}_{\tau }\in \ker (H_{\tau })$ such that

$$\begin{aligned} 0<\Vert {\varvec{x}}_{\tau }\Vert ^{m-1}_{\infty }\le \frac{\Vert H_\tau \Vert _2}{\gcd (H_\tau )} \le \sqrt{m}\Vert H_{\tau }\Vert _{\infty }\le \sqrt{m}\;h(P)\,. \end{aligned}$$

(18)

By the upper bound (17) we have

$$\begin{aligned} \Vert {\varvec{x}}_{\tau }\Vert _{\infty }\le r\,. \end{aligned}$$

Consequently, the lifted vector

$$\begin{aligned} \left( \begin{array}{c} {\varvec{x}}_{\tau }\\ {\varvec{0}}_{{{\bar{\tau }}}} \end{array} \right) \in C^n_m(r)\cap P\cap {\mathbb {Z}}^n, \end{aligned}$$

contradicting the assumption that $P\cap {\mathbb {Z}}^n$ is admissible for $C^n_m(r)$. The obtained contradiction completes the proof of Proposition 8.

Remark 2

A minor improvement of (18) can be obtained using a refinement of Siegel’s lemma proved in [1]. Further, the last inequality in (18) can be slightly strengthened using the following observation. Since $P\cap {\mathbb {Z}}^n$ is admissible for $C_m^n(1)$ and $m\ge 2$, we may assume that $H_{11}<H_{12}<\cdots <H_{1n}$. This allows choosing $H_\tau$ with $\Vert H_{\tau }\Vert \le \Vert H\Vert _{\infty }-n+m$.

6.1 Proof of Theorem 2

Take any $l, n\in {\mathbb {Z}}_{>0}$ with $1\le l <n/2$, $r\in {\mathbb {Z}}_{>0}$ and any $1\times n$ integer matrix $H\in {{\mathcal {R}}}(Q)$ with $Q=D(S^n_l(r))$. Consider the hyperplane $P=\ker (H)$. Set $m=2l$ and observe that

$$\begin{aligned} C^n_m(r)\cap {\mathbb {Z}}^n\subset Q\,. \end{aligned}$$

Therefore, the lattice $P \cap {\mathbb {Z}}^n$ is admissible for $C^n_m(r)$ and (12) implies (6).

7 Proof of Theorem 3

The proof of Theorem 3 is based on the following result, which is a special case of Proposition 1 (ii) in [2].

Proposition 12

Let S be an one-dimensional rational subspace of ${\mathbb {R}}^n$. When $1<m<n$, there is a rational subspace $T\supset S$ of dimension m in ${\mathbb {R}}^n$ with

$$\begin{aligned} \det (T\cap {\mathbb {Z}}^n)\le \gamma _{n-1,n-m}^{1/2} \det (S\cap {\mathbb {Z}}^n)^{(n-m)/(n-1)}\,. \end{aligned}$$

(19)

The constant $\gamma _{n-1,n-m}^{1/2}$ here is best possible.

Take any $Q\subset {\mathbb {Z}}^n$ and suppose that we are given an integer $1\times n$ matrix $H\in {{\mathcal {R}}}(Q)$. Let m be an integer with $1< m<n$ and let T be the subspace from Proposition 12, applied to the rational subspace S of ${\mathbb {R}}^n$ spanned by the row vector H.

By Theorem 11, applied with any $(n-m)\times n$ integer matrix M with $T=\ker (M)$, there exist m linearly independent integer vectors ${\varvec{g}}_1, \ldots , {\varvec{g}}_m\in T$ such that

$$\begin{aligned} \Vert {\varvec{g}}_1\Vert _{\infty } \cdots \Vert {\varvec{g}}_m\Vert _{\infty } \le \frac{\sqrt{\mathrm{det}(MM^{T})}}{\gcd (M)}=\det (T\cap {\mathbb {Z}}^n)\,. \end{aligned}$$

(20)

For a proof of the last equality in (20) we refer the reader to [18, Corollaries 5I-J]. Now we can form a matrix A with rows ${\varvec{g}}_1^\top , \ldots , {\varvec{g}}_m^\top$, so that $S\subset \mathrm {span}_{{\mathbb {R}}}(A^\top )$. Observe that $\ker (A)\subset \ker (H)$ and hence, $A\in {{\mathcal {R}}}(Q)$.

Finally, combining (20), (19) and the bound $\det (S\cap {\mathbb {Z}}^n)\le \sqrt{n}\Vert H\Vert _\infty$, we get the estimate (8):

$$\begin{aligned} \Vert A\Vert _{\infty }\le \gamma _{n-1,n-m}^{1/2} n^{(n-m)/(2(n-1))}\Vert H\Vert _\infty ^{(n-m)/(n-1)}\,. \end{aligned}$$