Before revision:  Doignon). We also present improvements on } the upper bounds for the Tverberg numbers of Z^3 and Z^j \times R^k  .
After revision:  Doignon). We also present improvements on } the upper bounds for the Tverberg numbers of Z^3 and Z^j \times R^k and an integer version of the well-known positive-fraction selection lemma of J. Pach .