An Erdös–Fuchs theorem for ordered representation functions
Letk=2 be a positive integer. We study concentration results for the ordered representationfunctionsr=k(A, n) = #{(a1= ··· =ak)¿ Ak:a1+···+ak=n}andr<k(A, n) = #{(a1<···<ak)¿ Ak:a1+···+ak=n}for any infinite set of non-negative integersA. Our main theorem is anErd ¿os–Fuchs-type result for bo...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2020 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/345088 |
| Acceso en línea: | https://hdl.handle.net/2117/345088 https://dx.doi.org/10.1007/s11139-020-00326-2 |
| Access Level: | acceso abierto |
| Palabra clave: | Additive Number Theory representation functions additive basis Erdös-Fuchs Theorem Classificació AMS::11 Number theory::11B Sequences and sets Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | Letk=2 be a positive integer. We study concentration results for the ordered representationfunctionsr=k(A, n) = #{(a1= ··· =ak)¿ Ak:a1+···+ak=n}andr<k(A, n) = #{(a1<···<ak)¿ Ak:a1+···+ak=n}for any infinite set of non-negative integersA. Our main theorem is anErd ¿os–Fuchs-type result for both functions: for anyc >0 and?¿{=, <}we show thatn¿j=0(r?k(A, j)-c)=o(n1/4log-1/2n)is not possible. We also show that the mean squared errorE?k,c(A, n) =1nn¿j=0(r?k(A, j)-c)2satisfies lim supn¿8E?k,c(A, n)>0. These results extend two theorems for the non-ordered representa-tion function proved by Erd ¿os and Fuchs in the case ofk= 2 (J. of the London Math. Society 1956) |
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