  We investigate the number $V_p(n)$ of distinct sites visited by an $n$-step
resetting random walker on a $d$-dimensional hypercubic lattice with resetting
probability $p$. In the case $p=0$, we recover the well-known result that the
average number of distinct sites grows for large $n$ as $\langle
V_0(n)\rangle\sim n^{d/2}$ for $d<2$ and as $\langle V_0(n)\rangle\sim n$ for
$d>2$. For $p>0$, we show that $\langle V_p(n)\rangle$ grows extremely slowly
as $\sim \left[\log(n)\right]^d$. We observe that the recurrence-transience
transition at $d=2$ for standard random walks (without resetting) disappears in
the presence of resetting. In the limit $p\to 0$, we compute the exact
crossover scaling function between the two regimes. In the one-dimensional
case, we derive analytically the full distribution of $V_p(n)$ in the limit of
large $n$. Moreover, for a one-dimensional random walker, we introduce a new
observable, which we call imbalance, that measures how much the visited region
is symmetric around the starting position. We analytically compute the full
distribution of the imbalance both for $p=0$ and for $p>0$. Our theoretical
results are verified by extensive numerical simulations.
