Dr. Arcidiacono was an expert witness for the plaintiffs at the trial. The study was written with economists at the University of Georgia and the University of Oklahoma.

A spokeswoman for Harvard, Rachael Dane, declined to comment on details of the study. But she said that the recruitment efforts it describes are a valuable source of students who are eventually admitted, accounting for 60 percent of freshmen in a typical year and more than 80 percent of minority freshmen.

For their recruitment drives, colleges buy names, ZIP codes, race and ethnicity and other information about students with test scores within a specified range from testing companies like the College Board and ACT. Students are often flooded with as many as 50 electronic solicitations via email, Snapchat and Instagram.

For the Class of 2018, Harvard sent out more than 114,000 letters and admitted 2,047 students. Almost half of those who qualified for a recruiting letter were members of underrepresented minorities.

Test-score cutoffs for the recruitment letters varied by race, gender and geography, and sometimes changed from year to year, according to testimony in the admissions trial. To get a letter in the fall of 2013, white and Asian-American men had to have scored at least 1380 on the SAT (converted from the equivalent on the PSAT), and black students and other underrepresented minorities had to have scored at least 1100.

While the numbers of applications Harvard gets from students of all races have increased over the years, the report said, African-American applications have soared, driven by students with lower SAT scores.

The increases were especially large for the classes applying between roughly 2003 and 2007, with the share of applicants who were black growing to 10.1 percent from 6.4 percent, the report said. But despite the growth in applications, the share of admitted students who were black stayed the same.