“lower part” or “foot of the mountain” - b’tachtit ha-har - means literally “under the mountain.” Thus the midrash that as they gathered there, G-d lifted up the entire mountain and held it over the Israelites' heads saying “If you accept the Torah, all well and good. If not, this is where you will be buried.”
Of course, as the rabbis note, this set up a problem: as R. Aha b. Jacob notes “This furnishes a strong protest against the Torah” – that acceptance was coerced.
Yet even so, they re-accepted it in the days of Ahasuerus, for it is written, [the Jews] confirmed, and took upon them [etc.]: (Esther 9:27)
[that is] they confirmed what they had accepted long before.
Nice answer. Made even more powerful when we look at more of the sentence:
The Jews confirmed, and took upon them, and upon their seed, and upon all who joined themselves to them. . .
It is an agreement binding on future generations – and explicitly includes converts! Clearly speaking to the realities of their time.
And I love that it is the story of Purim, that fairy-tale of the Diaspora with its grave dangers and unlikely opportunities, which ultimately confirms the agreement of the people to the covenantal relationship. G-d saves us from genocide and the response is a voluntary recommitment to that divine relationship throughout the generations.