You know a branch of your ancestors arrived in America about 1887. To learn if they ever became naturalized American citizens you would want to look for a petition to become a citizen and the completed naturalization papers. Count yourself lucky if that ancestor did not apply for citizenship until after 1906.
Your ancestor was not required to become a citizen. It was their own choice to seek naturalization. Prior to 1906, any of the written documents completed by an immigrant for citizenship was very basic information. Even what was requested by the government agencies varied by different states, counties and courts.
Starting in 1832 there was the declaration of intent which had the immigrant’s name, birth date and place and when they entered the United States, along with two witnesses' names. New laws for the Bureau of Immigration and Naturalized standardized the procedures. It took effect on September 27, 1906 and the documents and procedure became quite different for immigrants to process. First, the information submitted was on more uniform documents which had to be verified and match the records (ship manifest and entrance papers) of when they entered the country.
Each immigrant, even if they had arrived in the 1880s, had to file a Declaration of Intention or a Petition for Naturalization in a naturalization court. A Certification of Arrival of the person's entrance was needed. This requirement was especially true for those who arrived after June 29, 1906.
Each person applying for citizenship had to have a copy of the Declaration of Intention, Petition for Naturalization and a Certificate of Naturalization. Any declaration filed after September 27, 1906 were good for just seven years. Prior to 1906 there was no time limit. If the immigrate did not complete the necessary paper work within those seven years, the declaration expired and the person would have to start over again if they still wanted their American citizenship.
Some of the valuable information placed on the documents after September 1906 included full names (no nicknames), address, birthplace, nationality, address and country the person emigrated from, age, description of person (later also a photo was attached), marital status, name of spouse, port of entry, ship name, date of entry, children and their names, dates and birthplaces if they immigrated as well.
When all citizenship requirements were met a certification was issued to the naturalized citizen. The information the declarations of intention and the petitions provided did vary over the years after 1906, but it still surpassed in the amount and the substance of information compared to the much earlier documents for citizenship. So an ancestor applying for their citizenship from September 1906 or later would leave an extensive permanent record for future generations.