Sorry for jumping into these post.
Absence of one word/cognate in ancient Iranian and Sanskrit language does not mean that there is no interrelation among them.
If you talk about Aum/Om then we have Atman which has similar cognates in both PIE and Persian.
Etymology and meaning[edit]
"Ātman" (Atma, आत्मा, आत्मन्) is a Sanskrit word which means "essence, breath, soul."[7][8] It is related to PIE *etmen (a root meaning "breath"; cognates: Dutch adem, Old High German atum "breath," Modern German atmen "to breathe" and Atem "respiration, breath", Old English eþian).[7]
https://en.wikipedia.org/wiki/Ātman_(Hinduism)
Other examples that come to my mind.
New(in english)--->Nobo(in Bengali)---->Naya(in Hindi)
Except for linguistic arguments in support of AIT we have other textual arguments too. Since you are now reading all the ancient scriptures you may know that our forefathers referred foreigners as Yavanas/Mlecchas or barbarians. Yet they left Persians out of this tag. You should wonder why Persians were never considered as Mlecchas.
https://en.wikipedia.org/wiki/Mleccha
Further, there is evidence that Indians of the Vedic period actually had contact with people outside of the subcontinent, namely the Persians. The Persians, who ruled over the Indus river valley during this time (522-486 BC) were not designated as mleccha, perhaps because they did not interfere with the brahmanical way of life.[23]
(read the Territory sub para)
All south Indians were once considered as Mlecchas.(even the Bengali tribes and Kingdoms like Paundras/Vangas).
Also as per the archaeologists the Tamil script and the undecipherable MohenjoDaro Harrapa script bear remarkable similarity. How can you explain that?