TAOCP 3.3.4 Exercise 15
Let the hyperplanes defined by $U=(u_1,\ldots,u_t)$ be u_1x_1+\cdots+u_tx_t=h, where $h$ ranges over the integers.
Section 3.3.4: The Spectral Test
Exercise 15. ▶ [**] $[M30]$ Let $U$ be an integer vector satisfying (15). How many of the $(t-1)$-dimensional hyperplanes defined by $U$ intersect the unit hypercube ${(x_1, \ldots, x_t) \mid 0 \le x_j < 1 \text{ for } 1 \le j \le t}$? (This is approximately the number of hyperplanes in the family that will suffice to cover $L$.)
Verified: yes
Solve time: 1m14s
Let the hyperplanes defined by $U=(u_1,\ldots,u_t)$ be
$$ u_1x_1+\cdots+u_tx_t=h, $$
where $h$ ranges over the integers. Since $0\le x_j<1$ for all $j$, the quantity
$$ u_1x_1+\cdots+u_tx_t $$
ranges over the interval
$$ \sum_{u_j<0}u_j \le u_1x_1+\cdots+u_tx_t<\sum_{u_j>0}u_j. $$
Therefore a hyperplane intersects the unit hypercube if and only if its integer parameter $h$ satisfies
$$ \sum_{u_j<0}u_j \le h<\sum_{u_j>0}u_j. $$
The number of integers in this interval is
$$ \sum_{u_j>0}u_j-\sum_{u_j<0}u_j = \sum_{j=1}^t|u_j|. $$
Hence the number of $(t-1)$-dimensional hyperplanes that intersect the unit hypercube is
$$ \boxed{\sum_{j=1}^t|u_j|}. $$
∎