We create 4200 synthetic cities which vary in percent minority population and their residential segregation patterns. Of these, 1200 are modeled on existing cities, and 3000 are rectangular grid cities. In each city, we consider single-member voting district plans for a hypothetical city council election. A fair district plan is defined as one where the number of minority-majority districts is proportional to the city-wide minority population. Thus each city is summarized by three traits: minority percent, a measure of segregation, and availability of a fair district plan. We find that when the minority population is around 25%-33%, there is a positive correlation between the degree of segregation and the availability of proportional district plan. Consistently, when the minority population lives in a more diffuse residential pattern, there are fewer available proportional district plans. Finally, we develop a new method to validate runtime and sample size of an ensemble of district plans created by the GerryChain software program.

翻译：我们创建了4200个合成城市,这些城市的少数民族人口百分比及其居住隔离模式各不相同。 其中,1200个城市以现有城市为模范,3000个是长方格城市。我们考虑每个城市的假设市议会选举的单一成员投票区计划。一个公平的地区计划被定义为一个少数民族占多数的地区数量与整个城市的少数民族人口成比例的城市。因此,每个城市都有三个特点:少数民族比例、隔离程度和公平地区计划的可用性。我们发现,当少数民族人口约为25%至33%时,隔离程度和比例区计划的供应度之间就存在正相关关系。在少数民族人口以较为分散的居住模式生活时,可用的比例区计划就更少了。最后,我们开发了一种新的方法来验证Gerrychain软件方案所创建的地区计划的运行时间和抽样规模。