An electronic device may be provided with a sidewall, a display module separated from the sidewall by a gap, a display cover, a conductive bucket mounted to the display cover within the gap, and a phased antenna array mounted to the bucket for conveying millimeter wave signals through the display cover. The sidewall may form part of an antenna for conveying non-millimeter wave signals. The array may include resonating elements on a substrate. The resonating elements may be fed using feed terminals coupled to alternating sides of the resonating elements. Dielectric layers having a dielectric constant lower than that of the display cover may be provided on a surface of the display cover within the bucket. The array may operate with satisfactory efficiency despite the small amount of available space within the device, electromagnetic interference from the sidewall and the display module, and dielectric loading by the display cover.