So, in the past on here I've discussed the issues with the "Khazarian hypothesis", that Ashkenazi Jews are substantially descended from the Khazars, for its various issues. However, that doesn't necessarily mean that no Khazars integrated with the Jewish population of Eastern Europe--given what little we know of the Khazars, it might make some sense, since it seems that had interaction with other Jewish communities without a Khazar/non-Khazar divide. But there's still no evidence for it, besides the fact that Jews existed near Khazar areas before the beginning of mass migration from Central Europe--it just kinda makes things fit together alright. The basic model would be Khazar Jews founding Eastern European communities, particularly in modern-day Ukraine, which were then overwhelmed by immigrants from Central Europe, and assimilated into Ashkenazi Jewry. A similar pattern can be found in the fate of Spanish Jews--numerous and from a prestigious community, they immigrated to Middle Eastern communities and assimilated the existing community into them, rather than the other way round.
However, the population of "mountain Jews", a community which formerly lived in the Caucuses, is more likely than the bulk of European Jews to be descended from Khazars--the geography checks out, there's no convincing origin narrative to the contrary, and don't quote me on this but I think genetic evidence may indicate it too. Sadly, not nearly so much about them has been written as has been about Ashkenazim in general, but that is another potential place Khazar Jews could've ended up.
Keep in mind, too, that the number of Khazar Jews who would've been around is up for debate. It was once commonly held that the conversion of the Khazars was mostly legend, then that it was limited to ruling strata, then that it was first only the ruling classes but extended down to the common people with time. But the time elapsed between conversion to Judaism and the fall of Khazaria would not have been very large for many, so it's very conceivable that the conversion to Judaism would've been thought of perhaps in nominal terms, and discarded once the social circumstances that led to it were no longer present. The number who would've been more thoroughly Judaized might've been small enough that they could've assimilated into other Jewish communities (including Ashkenazim) relatively unnoticed, or small enough to not maintain a separate identity with the end of the Khazar social heirarchy.
Could you clarify what you're asking about David Al-Roy? There's no reason to think he was a Khazar. He was Iraqi, which had a large Jewish community. I've never seen any suggestion that an appreciable number of Khazars could've ended up in Iraq. None of the accounts of his life suggest anything of that sort, either.
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