Reynolds v. Sims
In Reynolds v. Sims, the U.S. Supreme Court required "one man, one vote" for all state legislative chambers, including state Senates, despite how the U.S. Senate is "one state, one vote." In other words, states much apportion both of its legislative chambers (Nebraska has only one) based on the population of the districts, so that a roughly equal number of people are represented by each legislator.
- [D]espite congressional approval of state legislative apportionment plans at the time of admission into the Union, even though deviating from the equal-population principle here enunciated, the Equal Protection Clause can and does require more. And an apportionment scheme in which both houses are based on population can hardly be considered as failing to satisfy the Guaranty Clause requirement. Congress presumably does not assume, in admitting States into the Union, to pass on all constitutional questions relating to the character of state governmental organization. In any event, congressional approval, however well-considered, could hardly validate an unconstitutional state legislative apportionment. Congress simply lacks the constitutional power to insulate States from attack with respect to alleged deprivations of individual constitutional rights.
Reynolds v. Sims, 377 U.S. 533, 582, 84 S. Ct. 1362, 1392 (1964).
This ruling was based on the Equal Protection Clause of the Fourteenth Amendment, interpreting it to require that all state legislative districts across states have roughly equal populations within them. Redistricting was required to comply with this ruling.