How to address an author showing replicated results from his paper?

Don't send your thesis unless requested. You may write to the author and say that you were interested to read his or her paper, that you applied the method to data from Argentina and you obtained similar results as described in the paper. You may give a little hint about any interesting quirks you found along the way but that is optional and if you do it, you must be very brief in this initial email. The author may be interested enough to write back enthusiastically, asking for more information, and then you can send a summary and your thesis. You may get a conference invite out of it. You may get an invitation to collaborate. You may get a reference to your thesis in this person's future work. You may get some new ideas for future work of your own. This email you will send will be a fun bit of icing on the cake for you, after a lot of hard work!

I have had a similar situation a few times.

In your thesis, it is crucial to cite where your method comes from and how you are applying it to your study.

As for writing to the academic, there is nothing wrong with letting them know that you have successfully applied their method to another dataset, but even for this and to ask for them to verify your results, it is very important to seek advice from your advisor first.