Afimoxifene in Reducing the Risk of Breast Cancer in Women With Mammographically Dense Breast

Inclusion Criteria:

* Women age 40-69 years, or less than 40 years if 5-year breast cancer Gail risk is ≥ 1.66%.
* Mammographically dense breast (heterogeneously dense \[C\] or extremely dense \[D\], based on American College of Radiology \[ACR\] BIRADS fifth edition classification or heterogeneously dense \[3\] or extremely dense \[4\], based on ACR BIRADS fourth edition classification) in either breast
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 (Karnofsky \>= 70%)
* White blood cells \>= 3,000/microliter
* Absolute neutrophil count \>= 1,500/microliter
* Platelets \>= 100,000/microliter
* Total bilirubin within normal institutional limits
* Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 1.5 × institutional upper limit of normal (ULN)
* Creatinine within normal institutional limits
* Participant must have a gynecology examination within the last 3 years, with no atypical hyperplasia and no cancer
* Premenopausal women taking non hormonal intra-uterine device (IUD) birth control method will be eligible, if they have been on the same IUD for at least 3 months prior to enrollment and plan to continue using the same method throughout the study
* Women of child-bearing potential must agree to use a reliable nonhormonal contraceptive method during the study and for 2 months after completing study medications; reliable nonhormonal methods of contraception include barrier contraception and an intra-uterine device (IUD); Note: women who had tubal ligation or had a partner who had undergone a vasectomy (and are monogamous) are eligible for the study and are not required to use barrier contraception
* If the participant is of childbearing potential, she must have a documented negative urine pregnancy test within 7 days prior to randomization
* Willingness to avoid exposing breast skin to natural or artificial sunlight (i.e. tanning beds) for the duration of the study
* Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

* History of allergic reactions attributed to compounds of similar chemical or biologic composition to 4-OHT gel
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, thromboembolic disease, or psychiatric illness/social situations that would limit compliance with study requirements
* Pregnant, or had given birth, or nursed at any time during the last 12 months
* Women with a previous history of invasive breast cancer or bilateral ductal carcinoma in situ (DCIS) or current untreated DCIS; women with a history of cancer within the last 3 years, except for non-melanoma skin cancer; women with unilateral DCIS (with or without radiation therapy) are eligible as long as they have an unaffected breast
* Prior bilateral breast surgery (mastectomy, segmental mastectomy, or breast augmentation surgery including breast implants or breast reductions)
* Women with "mosaic mammographic screening views", i.e., whose larger breast size precludes being imaged within a single mammographic screening view
* Women with active liver disease, abnormal uterine bleeding, or prior diagnosis of endometrial hyperplasia
* Prior use of selective estrogen receptor modulators (SERMS) and aromatase inhibitors (AIs) for prevention or therapy
* Skin lesions on the breast that disrupt the stratum corneum (e.g., eczema, ulceration)
* Treatment with any investigational drug or investigational biologic within 30 days of initiating study treatment or during the study