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In the projective space $\mathrm{PG}(N,q)$ over the Galois field of order$q$, $N\ge3$, an iterative step-by-step construction of complete caps by addinga new point on every step is considered. It is proved that uncovered points areevenly placed on the space. A natural conjecture on an estimate of the numberof new covered points on every step is done. For a part of the iterativeprocess, this estimate is proved rigorously. Under the conjecture mentioned,new upper bounds on the smallest size $t_{2}(N,q)$ of a complete cap in$\mathrm{PG}(N,q)$ are obtained, in particular, \begin{align*}t_{2}(N,q)<\frac{\sqrt{q^{N+1}}}{q-1}\left(\sqrt{(N+1)\lnq}+1\right)+2\thicksim q^\frac{N-1}{2}\sqrt{(N+1)\ln q},\quad N\ge3.\end{align*} A connection with the Birthday problem is noted. The effectivenessof the new bounds is illustrated by comparison with sizes of complete capsobtained by computer in wide regions of $q$.

Upper bounds on the smallest size of a complete cap in $\mathrm{PG}(N,q)$, $N\ge3$, under a certain probabilistic conjecture