An electronic device may be provided with wireless communications circuitry and control circuitry. The wireless communications circuitry may include centimeter and millimeter wave transceiver circuitry and a phased antenna array. A dielectric cover may be formed over the phased antenna array. The phased antenna array may transmit and receive antenna signals through the dielectric cover. The dielectric cover may have a surface that faces the phased antenna array and may have a curvature. The antenna elements of the phased antenna array may be formed on a dielectric substrate. The dielectric substrate may have one or more thinned regions between antenna elements of the phased antenna array to reduce surface wave interference between adjacent antennas. The dielectric substrate may have a smaller thickness in the thinned region than in the regions under the antenna elements. The dielectric substrate may be totally removed in the thinned region.