"But thou, Bethlehem Ephratah, though thou be little among the thousands of Judah, yet out of thee shall he come forth unto me that is to be ruler in Israel; whose goings forth have been from of old, from everlasting." Micah 5:2
"In that day will I raise up the tabernacle of David that is fallen, and close up the breaches thereof; and I will raise up his ruins, and I will build it as in the days of old:
12. That they may possess the remnant of Edom, and of all the heathen, which are called by my name, saith the LORD that doeth this." Amos 9: 11-12
"Afterward shall the children of Israel return, and seek the LORD their God, and David their king; and shall fear the LORD and his goodness in the latter days," Hosea 3: 5
"And I will bring them (Israel), and they shall dwell in the midst of Jerusalem: and they shall be my people, and I will be their God, in truth and in righteousness...And it shall come to pass, that as ye were a curse among the heathen, O house of Judah, and house of Israel; so will I save you, and ye shall be a blessing...In those days it shall come to pass, that ten men shall take hold out of all languages of the nations, even shall take hold of the skirt of him that is a Jew, saying, We will go with you: for we have heard that God is with you" Zechariah 8:8,13: 23.
In Acts 15: 14-17 even James, who bragged to Paul in Acts 21: 19-21 that there were thousands of Jews who believe and are zealous of the law, said "Simeon hath declared how God at the first did visit the Gentiles, to take out of them a people for his name.
15. And to this agree the words of the prophets; as it is written,
16. After this I will return, and will build again the tabernacle of David, which is fallen down; and I will build again the ruins thereof, and I will set it up:
17. That the residue of men might seek after the Lord, and all the Gentiles, upon whom my name is called, saith the Lord, who doeth all these things.
James is quoting Amos 9: 11-12.
All of the four texts quoted above are about the restoration of Israel. But James says that Amos 9: 11-12 agrees with Peter's statement that Gentiles heard the word of the Gospel and believed, referring to Acts 10. The restoration of Israel according to James turned out to be one in which God led some of the Gentiles to believe on Christ. He could have added that this fulfills Hosea 2: 23.
And, what is important, James adds in Acts 15: 15 that not only does Amos 9: 11-12 agree with God's making Gentiles his people, but that the prophets, plural, not just Amos, agree with this. This means that James interprets other Old Testament prophecies of the restoration of Israel as being about a restoration which turns out to the one in which Christ appears and brings in the New Covenant.