Oral Tamoxifen vs. TamGel vs. Control in Women With Atypical Hyperplasia, Lobular Carcinoma In Situ, or Increased Breast Cancer Risk

Inclusion Criteria: * Willing to return to enrolling institution for follow-up * Willing to complete required testing * Ability to complete questionnaire by themselves or with assistance * Female (sex that was assigned at birth) * Ipsilateral intact breast with histology confirmation of atypical ductal or lobular hyperplasia, or lobular carcinoma in situ (LCIS), within the last 12 months, whether surgically excised or not.; OR neither AH nor LCIS but increased breast cancer risk defined as either: * Gail model (Breast Cancer Risk Assessment Tool) 5 year breast cancer risk of \>= 3%, or * International Breast Intervention Study model 10 year breast cancer risk of \>= 5%. * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 (Karnofsky \>= 70%) * The effects of topical afimoxifene (4-OHT) gel on the developing human fetus at the recommended therapeutic dose are unknown. However, oral tamoxifen is Pregnancy Category D-positive evidence of human fetal risk. For this reason, and because triphenylethylene antiestrogens, including tamoxifen, are known to be teratogenic, women of childbearing potential and their male partners must agree to use at least one effective form of birth control (abstinence is not an allowed method) prior to study entry and for the duration of study participation, and for 2 months following the last dose of study medications (participant can resume oral birth control pills for effective birth control measures after post-treatment biopsy …