How could we make the Kingdom of Arles, South of France be incorporated to France in the Dark Ages.
I was thinking if Bertha of Burgundy was able to sire a son with Robert II, probably we would have Arles integrated to France sooner than in OTL.That will be difficult, your objective would be to not have the Holy Roman Emperor, Conrad II Salian, inherit Arles when Rudolph III dies in 1032. Alternatively, since Rudolph III was essentially a weak willed monarch and was pressed by both Conrad and his predecessor Henry II into making them his heir, perhaps Robert II could do the same (Henri I of France was more realistic and would probably not be tied down with conflicts abroad with his demense being small).
Long story short, either have Robert II push for a claim or keep the Welfs Arletian line going long enough.
And perhaps that would mean France also control Savoy and Aosta..If France manages to gain the Arelat, then the situation in Italy also could become a bit more interesting too.
I also agree, that if Robert II and Bertha would have had a son and heir that he either could push his claim or alternatively convince Rudolph III to name him as his heir.