We will be using the following lemmas:
Lemma 6.4 (from Probability in High Dimension): For zero-mean subgaussian random variables $Y_1, ..., Y_k$, a constant $c$ exists such that $$E\left[\max_{i \leq k} Y_i\right] \leq c \max_i |Y_i|_{\text{subgaussian}} \sqrt{\log k}$$
Lemma 6.19 (from Probability in High Dimension): For $(\epsilon^2)$-subgaussian random variables $Y_1, ..., Y_k$ and any $t > 0$, $$P\left(\max_{1 \leq i \leq k} Y_i \geq t \right) \leq 2k\exp\left(-\frac{t^2}{2\epsilon^2}\right)$$
Consider $Z_1, ..., Z_n$ as independent standard normal random variables and $X_k$ as binary vectors of length $n$ with $k$ ones, each with unit Euclidean norm. For any $X \in X_k$, the variable $Z^T X$ is Gaussian with zero mean, implying $E[Z^T X] = 0$. The distribution of $Z^T X$ remains Gaussian due to the properties of Gaussian distributions, where the linear combination of Gaussian variables is also Gaussian. Given that each component of $Z$ is standard normal and $X$ has a unit norm, the variance of $Z^T X$ is 1, making it a standard normal variable.
To apply the subgaussian norm definition, $|Y|_{\psi_2} = \inf \{ t > 0 : E[\exp(Y^2 / t^2)] \leq 2 \}$, we note that for standard Gaussian variables, this norm is a constant factor. Specifically, for a standard normal variable, the subgaussian norm can be directly related to its variance, leading to the identification of $\sigma^2 = 1$ for $Z^T X$. This quantifies the tail behavior, with its value being 1 reflecting the standard deviation of the Gaussian distribution.
Using Lemma 6.4, the expectation of the maximum deviation of $Z^T X$ from zero, for any $X$ in the set $X_k$, is bounded as follows: $$E\left[\max_{X \in X_k} \left|Z^T X\right|\right] \leq c \sigma \sqrt{\log k}$$ where $c$ is a constant for zero-mean subgaussian random variables, as established in Lemma 6.4, and $\sigma = 1$ in our case. This equation provides an upper limit on the expected maximum deviation, using the logarithmic relationship with $k$ to illustrate how the expected maximum grows as the number of vectors in $X_k$ increases.
Using Lemma 6.19, we derive the tail probability bound for the maximum deviation of $Z^T X$ exceeding a threshold $t$: $$P\left(\max_{X \in X_k} \left|Z^T X\right| \geq t \right) \leq 2k\exp(-t^2/(2\sigma^2))$$ Here, $\sigma^2 = 1$ represents the squared subgaussian norm of $Z^T X$, and highlights the rapid decrease in the probability of observing extreme deviations as $t$ increases.