One news item that recently made waves came from a Facebook post by Ahmed Ben Naoum, a professor of sociology at the University of Perpignan, who, as reported by El Watan, insists that the Chaamba (properly šʕanba) are not Arabs but rather Zenati Berbers. The ancestry of the Chaamba is not something I can comment on professionally - if that mattered, which it shouldn't, a look at their Y-chromosomes would be the way to go. Nor can I say much on their historical self-identification, though at present it's extremely clear that the Chaamba consider themselves Arab (more specifically, a branch of Banu Sulaym). However, the article also touches on their language:
«Les Cha’anba font partie de la majorité zénète de ce pays. Ils n’ont aucun mythe fondateur les rattachant aux ‘‘Arabes’’ ! Eux-mêmes ont été arabisés comme l’ont été les autres Zénètes, sauf à dire qu’ils expriment leur culture dans une des langues arabes qu’ils ont largement ‘‘zénétisée’’ dans la morphologie et la syntaxe.»
[The Sha'anba are part of the Zenati majority of this country. They have no foundation myth attaching them to the "Arabs"! They themselves have been Arabised like the other Zenatis, but that is only to say that they express their culture in one of the Arabic languages which they have extensively "Zenatified" in morphology and syntax.]
This is not correct. The dialect of the Chaamba is one of the few dialects of the Algerian Sahara for which a grammatical description has been published (Grand'Henry 1976), and its morphology, at least, is pretty well studied. Judging by this material, there is no discernible Zenata (or other Berber) influence on the morphology or syntax of the dialect at all. In this respect, it agrees with Algerian Arabic more generally. Very few dialects of Algerian Arabic show significant morphological influence from Berber; only a few areas, such as Jijel or Adrar, even have Berber plurals for nouns borrowed from Berber, and no dialect anywhere is reported to has borrowed Berber verbal morphology. Many dialects have a few abstract nouns in ta-...-t - usually with negative meanings - but this formation is hardly productive. Syntactic influence is plausible a priori, but has not been adequately demonstrated anywhere in Algerian Arabic (except Jijel), much less for the dialect of the Chaamba.
A better place to look for Berber influence in Algerian dialects, generally speaking, is phonology and vocabulary. In phonology, the phoneme ẓ and the merger of the short vowels can both plausibly - although not certainly - be attributed to Berber influence; however, it is unclear from Grand'Henry's rather poor description of the phonology whether even these apply in the Chaamba dialect. The vocabulary listed by Grand'Henry includes very few Berber loans, and most of the latter are pan-Algerian, eg həžžala "widow", atay "tea" (the latter ultimately from Chinese); the only rarer ones noted are two types of date, taqərbŭšt and tantmŭšt, which would naturally be easily borrowed from Berber-speaking oasis dwellers. On the basis of the available data, it's safe to say that the Zenati influence in the dialect of the Chaamba, like the Zenati influence in most Algerian dialects whether spoken by people of Berber ancestry or not, is very limited. It would be very interesting to study the extent of Berber influence in the Arabic spoken in different regions of Algeria, and how it varies. But such a study should not be expected to provide proof that Algerians in general, or any specific group of Algerians in particular, are of Amazigh ancestry. If for some reason you want to know about ancestry, ask a geneticist, not a linguist (nor, I would suggest, a sociologist).