The time of giving our Torah is not explicitly written, and the connection between it and Shavuot needs clarification and incandescence, since even Shavuot is not set for a day in the month, and its time is on the fiftieth day of Nisan, the day of the Omer sacrifice. 'Seder Olam' (accepted on all issues in the Gemara), To 10 in the month of Nissan was the Great Shabbat, the 16th day of Nisan was Friday on Shabbat and the Torah was given on Shabbat - after all, the giving of Torah was on the NA on the day of the sacrifice of Omer and not on Shavuot.
The Gemara in Tractate Shabbat (Page 87 Amud 1) discusses at length the question of which day of the month the Torah was given, thus bringing about a dispute between R. Yossi, who claims that the Torah was given on the 7th of Sivan, and Rabbanan, who believe it was given on the 6th of Sivan.
By all accounts, the days of restriction began on the 4th of Sivan. The controversy is whether the words of God were fulfilled - "for on the third day the Lord came down before all the people on Mount Sinai" (Shmot 19:11), and then the Torah was given on Friday in Sivan, or whether the words of Moses were fulfilled - "were ready for three days" (19:15), and then the Torah was given precisely on the seventh of it...
He who looks at the simplifications of the readings seems to conclude that the Torah was given on the third of Sivan. For "in the third month ... on this day came the Sinai Desert" (19:1), and immediately Moses ascended to the Lord. , On the 3rd of Sivan - the Lord came down on Mount Sinai and said the Ten Commandments.
Sages in their midrashim addressed this problem, and 'filled' the first three days of Sivan with various events. On the first day, Israel rested from the toil of the road, on the second they were told "and you will be my kingdom of priests" Tractate Shabbat (Page 87 Amud 1). The question that arises from this is clear: what did the sages see to extend the time until the giving of the Torah, to give a day (not mentioned) to rest and split the rest of the speech into different days, and not to "compress" things, As a simplification of readings, for the first three days of the month of Sivan?
The need of the sages to postpone the giving of the Torah until the Lord in Sivan may stem from what is said in chapter Shmot 24:16. Indeed, the commentators there were divided as to whether these six days are the six days before the giving of the Torah (according to Ramban and Rasag (r. Saadia Gaon)), or the first six days out of the forty after the giving of the Torah (Avraham Ibn Ezra).