We consider a discrete-time process adapted to some filtration which lives on a (typically countable) subset of ℝd, d≥2. For this process, we assume that it has uniformly bounded jumps, and is uniformly elliptic (can advance by at least some fixed amount with respect to any direction, with uniformly positive probability). Also, we assume that the projection of this process on some fixed vector is a submartingale, and that a stronger additional condition on the direction of the drift holds (this condition does not exclude that the drift could be equal to 0 or be arbitrarily small). The main result is that with very high probability the number of visits to any fixed site by time n is less than n1/2-δ for some δ > 0. This in its turn implies that the number of different sites visited by the process by time n should be at least n1/2+δ. © 2012 Springer Science+Business Media, LLC.272601617Alves, O.S.M., Machado, F.P., Popov, S.Y., The shape theorem for the frog model (2002) Ann. Appl. Probab., 12 (2), pp. 534-547Aspandiiarov, S., Iasnogorodski, R., Menshikov, M., Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant (1996) Ann. Probab., 24 (2), pp. 932-960Benjamini, I., Izkovsky, R., Kesten, H., On the range of the simple random walk bridge on groups (2007) Electronic J. Probab., 12 (20), pp. 591-612Benjamini, I., Wilson, D.B., Excited random walk (2003) Electron. Commun. Probab., 8, pp. 86-92Bérard, J., Ramírez, A., Central limit theorem for the excited random walk in dimension d≥2 (2007) Electron. Commun. Probab., 12, pp. 303-314Černý, J., Moments and distribution of the local time of a two-dimensional random walk (2007) Stoch. Process. Appl., 117 (2), pp. 262-270Csáki, E., Csörgö, M., Földes, A., Révész, P., On the local time of random walk on the 2-dimensional comb (2011) Stoch. Process. Appl., 121 (6), pp. 1290-1314Csáki, E., Földes, A., Révész, P., Joint asymptotic behavior of local and occupation times of random walk in higher dimension (2007) Studia Sci. Math. Hung., 44 (4), pp. 535-563Csáki, E., Révész, P., Rosen, J., Functional laws of the iterated logarithm for local times of recurrent random walks on ℤ2 (1998) Ann. Inst. Henri Poincaré B, Probab. Stat., 34 (4), pp. 545-563Donsker, M.D., Varadhan, S.R.S., On the number of distinct sites visited by a random walk (1979) Commun. Pure Appl. Math., 32 (6), pp. 721-747Hamana, Y., An almost sure invariance principle for the range of random walks (1998) Stoch. Process. Appl., 78 (2), pp. 131-143van der Hofstad, R., Holmes, M.P., Monotonicity for excited random walk in high dimensions (2010) Probab. Theory Relat. Fields, 147 (1-2), pp. 333-348Hughes, B.D., (1995) Random Walks and Random Environments, 1. , Oxford: ClarendonMacPhee, I.M., Menshikov, M.V., Wade, A.R., Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts (2012) J. Theor. Probab., , arXiv: 0806. 4561Marcus, M.B., Rosen, J., Logarithmic averages for the local times of recurrent random walks and Lévy processes (1995) Stoch. Process. Appl., 59 (2), pp. 175-184Menshikov, M., Popov, S., Ramírez, A., Vachkovskaia, M., On a general many-dimensional excited random walk (2012) Ann. Probab., , arXiv: 1001. 1741Rau, C., Sur le nombre de points visités par une marche aléatoire sur un amas infini de percolation (2007) Bull. Soc. Math. Fr., 135 (1), pp. 135-169Spitzer, F., (1976) Principles of Random Walks, , 2nd edn., Berlin: SpringerZerner, M.P.W., Recurrence and transience of excited random walks on ℤd and strips (2006) Electron. Commun. Probab., 11, pp. 118-128