Thank you for your thoughts.
I’m trying to figure out, if it was possible for him to be in United States for five consecutive years. He left Denmark the first time June 1880 through Hamburg, was back in Denmark Feb 1885, he was present in both census. Did not find him in Feb 1890 census, that doesn’t mean he wasn’t in Denmark at the time, next census is Feb 1901 where he lived with his father. That is a window of 15 years he could have been in the US, but there are not records of him going back to the US until 1902, not in the Danish Emigration Record database you are referring to, nor in ancestry.com for passenger lists.
If he in fact was in United States he could have arrived approx 1886, been there 5 years, which would make it approx 1891, before he could file the “Declaration of Intention to Become a US Citizen”, add 3 years before the citizenship is granted. The period to look for citizenship would then be 1894-1900.
If the record you found is him, then he did lie about being a US Citizen in 1902, you don’t apply again in 1911, if you already got citizenship. The record you found is for a H. A. Pederson, Cir. Ct. Kane Co, Ill, that is all the information stated on the record. It still could be him, but I don’t really know how to be sure.
So if he told the truth a citizenship would have been granted between 1894-1900, well that narrows it down a bit, I just think it is weird I can’t find him anywhere after 1902 in the United States.