Maybe some other volunteer researcher will find the answer.
An immigrant must live in the USA continuously for five years before he/she can become a citizen. The immigrant must file a "Declaration of Intention To Become a US Citizen" three years before citizenship is granted. This rule was in effect also at the times Holger A. Petersen came to the USA.
I don't know why he would lie about having citizenship papers, as he could be admitted to the USA as a non-citizen also. Some persons were excluded from immigrating, such as the feeble-minded or people ill with tuberculosis or those who might be unable to support themselves. Such persons would be admitted if they were already US citizens of course.
Holger was a cook, a single man of middle age. No obstacle to his entry is noted on the Ellis Island passenger manifest.
On the Ellis Island passenger manifest, he gives his final destination as New Haven, Connecticut. I did not find his declaration of intention or his naturalization papers in Connecticut.
Danish Emigration Record 1902
Navn: Petersen, Holger Amandus Stilling: Kok
Alder: 43 Bestemmelsessted: N.Y.
Kontrakt nr.: 259800 Forevisningsdato: 9/24/1902
Fødested: Kjøbh. Fødesogn: København
Sidste oph.sogn: ? Sidste oph. amt: - USA
Sidste oph.sted: U.S.A. Bestemmelses land: USA
Bestemmelses by: New York City Bestemmelses stat: New York State
I don't know if this is your H.A. Petersen, below; I have not been able to open this file.
U.S. Subject Index to Correspondence and Case Files of the Immigration and Naturalization Service (from database at Ancestry.com)
Names of Persons in Entry: H A Peterson
Subject Title: Declarations Of Intentions Applications For Copies
Date: Jan 1911
INS Reference Number: 117,692