RESTRICTED PARTITION FUNCTIONS AND ADDITIVE COMPLEMENTS

Source abstract

Abstract Let N ${\mathbb N}$ double struck upper N be the set of positive integers. For subsets A , M ⊆ N $\mathcal {A},\mathcal {M}\subseteq {\mathbb N}$ script upper A comma script upper M subset of or equal to double struck upper N and n ∈ N $n\in {\mathbb N}$ n element of double struck upper N , let p ( n , A , M ) $p(n,\mathcal {A},\mathcal {M})$ p left parenthesis n comma script upper A comma script upper M right parenthesis denote the number of representations of n in the form n = ∑ a ∈ A m a a $n=\sum _{a\in \mathcal {A}}m_a a$ n equals sigma summation Underscript a element of script upper A Endscripts m Subscript a Baseline a , where m a ∈ M ∪ { 0 } $m_a\in \mathcal {M}\cup \{0\}$ m Subscript a Baseline element of script upper M union StartSet 0 EndSet for all a ∈ A $a\in \mathcal {A}$ a element of script upper A and only finitely many m a $m_a$ m Subscript a are nonzero. We prove that there exist two infinite sets A = { a n } n = 1 ∞ $\mathcal {A}=\{a_n\}_{n=1}^{\infty }$ script upper A equals left brace a Subscript n Baseline right brace Subscript n equals 1 Superscript infinity and M $\mathcal {M}$ script upper M of positive integers such that lim n → ∞ ( ( log ⁡ a n + 1 − log ⁡ a n ) / log ⁡ n ) = + ∞ $\lim _{n\to \infty }(({\log a_{n+1}-\log a_n})/{\log n})=+\infty$ limit Underscript n right arrow infinity Endscripts left parenthesis left parenthesis log a Subscript n plus 1 Baseline minus log a Subscript n Baseline right parenthesis divided by log n right parenthesis equals plus infinity , p ( n , A , M ) > 0 $p(n,\mathcal {A},\mathcal {M})>0$ p left parenthesis n comma script upper A comma script upper M right parenthesis greater than 0 for every n ∈ N $n\in {\mathbb N}$ n element of double struck upper N and p has polynomial growth. More generally, we provide a construction that associates restricted partition functions of polynomial growth with additive complements satisfying a simple counting condition. This answers a question of Dai and Chen [‘On two problems of Ljujić and Nathanson’, C. R. Math. Acad. Sci. Paris 354 (2016), 235–238] in the affirmative.