According to sources cited in Seder Hadorot, Tevet 28 is both the birthday and the day of passing of Shimon the son of Jacob; other sources place the date as Tevet 21. (See the entry for Tevet 21).
Shimon ben Shetach successfully completed the expulsion of the Sadducees (a sect which denied the Oral Torah and the authority of the Sages) who had dominated the Sanhedrin (Supreme Court), replacing them with his Torah-loyal disciples, on the 28th of Tevet of the year 3680 from creation (81 BCE).
Rebbetzin Chana Schneerson (1879 (O.S.) - 1964), mother of the Lubavitcher Rebbe, was born on Tevet 28.
What does fasting on Yom Kippur achieve?
That even our physical bodies can feel the precious closeness of this day.
Yom Kippur is such a holy day, the body receives life from fasting.