An $n \times n$ matrix with $\pm 1$ entries which acts on $\mathbb{R}^n$ as a scaled isometry is called Hadamard. Such matrices exist in some, but not all dimensions. Combining number-theoretic and probabilistic tools we construct matrices with $\pm 1$ entries which act as approximate scaled isometries in $\mathbb{R}^n$ for all $n$. More precisely, the matrices we construct have condition numbers bounded by a constant independent of $n$. Using this construction, we establish a phase transition for the probability that a random frame contains a Riesz basis. Namely, we show that a random frame in $\mathbb{R}^n$ formed by $N$ vectors with independent identically distributed coordinates having a non-degenerate symmetric distribution contains many Riesz bases with high probability provided that $N \ge \exp(Cn)$. On the other hand, we prove that if the entries are subgaussian, then a random frame fails to contain a Riesz basis with probability close to $1$ whenever $N \le \exp(cn)$, where $c<C$ are constants depending on the distribution of the entries.

翻译：一个$n \times n$的矩阵，其元素为$\pm 1$，且在$\mathbb{R}^n$上作用为缩放等距变换，称为哈达玛矩阵。这类矩阵存在于部分维度中，但并非所有维度。结合数论与概率论工具，我们构造了元素为$\pm 1$的矩阵，这些矩阵在所有$n$的$\mathbb{R}^n$中均能作为近似缩放等距变换。更精确地说，我们构造的矩阵条件数被一个与$n$无关的常数所界。利用这一构造，我们建立了随机框架包含Riesz基概率的相变现象。具体而言，我们证明：由$N$个独立同分布坐标向量（服从非退化对称分布）在$\mathbb{R}^n$中形成的随机框架，当$N \ge \exp(Cn)$时，以高概率包含大量Riesz基；另一方面，若坐标满足亚高斯分布，当$N \le \exp(cn)$时（其中$c<C$为依赖于分布常数的参数），该随机框架以接近1的概率不包含任何Riesz基。