1-sparse recovery algorithm
Without the mod p, the running time to check the condition would be exponential, which is unacceptable.  Also, it would require exponential space to store $w_3$ and to check the condition.
Also, the procedure involves choosing an integer $z$ uniformly at random from $\&gt;0,\dots,p-1\}$.  You can choose an integer uniformly at random from $\{0,\dots,p-1\}$, but you can't choose it uniformly at random from the set of all integers.  So, you'd have to think about exactly what alternative approach you had in mind if you wanted to omit the mod p.