On function $SX$ of additive complements

Two infinite sequences $A$ and $B$ of non-negative integers are called additive complements if their sum contains all sufficiently large integers. Let $A(x)$ and $B(x)$ be the counting functions of $A$ and $B$ and let $\limsup \limits _{x\rightarrow \infty }A(x)B(x)/ x$ $=\alpha (A, B)$. Recently, the authors [Proceedings of the American Mathematical Society 138 (2010), 1923-1927] proved that for additive complements $A$ and $B$, if $\alpha (A, B)<5/4$ or $\alpha (A, B)>2$, then $A(x)B(x)-x\rightarrow +\infty$ as $x\to \infty$. In this paper, we prove that for any $\varepsilon >0$ there exist additive complements $A$ and $B$ with $2-\varepsilon <\alpha (A, B) <2$ and $A(x)B(x)-x=1$ for infinitely many positive integers $x$.

On additive complements. IV

For two infinite sequences A and B of non-negative integers, if every sufficiently large integer can be expressed as the sum of two elements taken from A and B, then we call such A, B additive complements. Let A(x) (resp. B(x)) be the counting functions of A (resp. B). Motivated by a result of Sárközy and Szemerédi, the authors proved that if limsupA(x)B(x)/x&lt;3−3 or limsupA(x)B(x)/x&gt;2, then A(x)B(x)−x→+∞ as x→+∞. Afterwards we also posed a natural conjecture that: for additive complements A, B, there exists a set T of non-negative integers with density one such that A(x)B(x)−x→+∞asx∈Tandx→+∞. In this paper, we confirm this conjecture. Furthermore, we obtain a stronger evaluation for the special additive complements.

Let A and B be two subsets of the nonnegative integers. We call A and B additive complements if all sufficiently large integers n can be written as a+b, where a∈A and b∈B. Let S={1 2 ,2 2 ,3 2 ,···} be the set of all square numbers. Ben Green was interested in the additive complement of S. He asked whether there is an additive complement B={b n } n=1 ∞ ⊆ℕ which satisfies b n =π 2 16n 2 +o(n 2 ). Recently, Chen and Fang proved that if B is such an additive complement, thenlim sup n→∞ π 2 16n 2 -b n n 1/2 logn≥2 π1 log4.They further conjectured thatlim sup n→∞ π 2 16n 2 -b n n 1/2 logn=+∞.In this paper, we confirm this conjecture by giving a much more stronger result, i.e.,lim sup n→∞ π 2 16n 2 -b n n≥π 4.

Two sequences A and B of non-negative integers are called additive complements, if their sum contains all suffciently large integers. Let A(x) and B(x) be the counting functions of A and B, respectively. In 1994, Sarkozy and Szemeredi proved that, for additive complements A and B, if limsup A(x)B(x)=x ≤ 1, then A(x)B(x)-x→+∞ as x→+∞. In this paper, motivated by a recent result of Ruzsa, we prove the following result: for additive complements A, B with Narkiewicz's condition: A(2x)=A(x)→ as x→+∞, we have A(x)B(x)-x>(1+o(1))a(x)=A(x) as x→+∞, where a(x) is the largest element in A⋂[1,x]. Furthermore, this is the best possible. As a corollary, for additive complements A, B with Narkiewicz's condition: A(2x)=A(x)→1 as x→+∞ and any M >1, we have A(x)B(x)-x>A(x)M for all suffciently large x.

On additive complements. III

On finite additive complements

Additive complements of the squares

On minimal additive complements of integers

Two sets $A,B$ of positive integers are called exact additive complements if $A+B$ contains all sufficiently large integers and $A(x)B(x)/x\rightarrow 1$ . For $A=\{a_1&lt;a_2&lt;\cdots \}$ , let $A(x)$ denote the counting function of A and let $a^*(x)$ denote the largest element in $A\bigcap [1,x]$ . Following the work of Ruzsa [‘Exact additive complements’, Quart. J. Math.68 (2017) 227–235] and Chen and Fang [‘Additive complements with Narkiewicz’s condition’, Combinatorica39 (2019), 813–823], we prove that, for exact additive complements $A,B$ with ${a_{n+1}}/ {na_n}\rightarrow \infty $ , $$ \begin{align*}A(x)B(x)-x\geqslant \frac{a^*(x)}{A(x)}+o\bigg(\frac{a^*(x)}{A(x)^2}\bigg) \quad\mbox{as } x\rightarrow +\infty.\end{align*} $$ We also construct exact additive complements $A,B$ with ${a_{n+1}}/{na_n}\rightarrow \infty $ such that $$ \begin{align*}A(x)B(x)-x\leqslant \frac{a^*(x)}{A(x)}+(1+o(1))\bigg(\frac{a^*(x)}{A(x)^2}\bigg)\end{align*} $$ for infinitely many positive integers x.

Two infinite sets $A$ and $B$ of nonnegative integers are called additive complements if their sumset contains every nonnegative integer. In 1964, Danzer constructed infinite additive complements $A$ and $B$ with $A(x)B(x) = (1 + o(1))x$ as $x \rightarrow \infty$, where $A(x)$ and $B(x)$ denote the counting function of the sets $A$ and $B$, respectively. In this paper we solve a problem of Chen and Fang by extending the construction of Danzer.

On additive complement of a finite set

Let f ( n , k , s ) f(n,k,s) denote the cardinality of the smallest set A A of nonnegative k k -th powers such that every integer in [ 0 , n ] [0,n] is a sum of s s elements of A A , and let β ( k , s ) = lim su p n → ∞ log ⁡ f ( n , k , s ) / log ⁡ n \beta (k,s) = {\text {lim su}}{{\text {p}}_{n \to \infty }}\log f(n,k,s)/\log n . Clearly, β ( k , s ) ⩾ 1 / s \beta (k,s) \geqslant 1/s . In this paper it is proved that f ( n , k , s ) &gt; c n 1 / ( s − g ( k ) + k ) f(n,k,s){\text { &gt; }}c{n^{1/(s - g(k) + k)}} for all n ⩾ n 1 ( k , s ) n \geqslant {n_1}(k,s) , where g ( k ) g(k) is defined as in Waring’s problem, and β ( k , s ) ∼ 1 / s \beta (k,s) \sim 1/s as s → ∞ s \to \infty .

Two infinite sets A and B of non-negative integers are called perfect additive complements of non-negative integers, if every non-negative integer can be uniquely expressed as the sum of elements from A and B. In this paper, we define a Lagrange-like spectrum of the perfect additive complements (L for short). As a main result, we obtain the smallest accumulation point of the set L and prove that the set L is closed. Other related results and problems are also contained.

In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous Ramanujan form x 2 + y 2 + 1Oz 2 ; equivalently the form 2x 2 + 5y 2 +4T z represents all integers greater than 1359, where T z denotes the triangular number z(z + 1)/2. Given positive integers a, b, c we employ modular forms and the theory of quadratic forms to determine completely when the general form ax 2 + by 2 + cT z represents sufficiently large integers and to establish similar results for the forms ax 2 + bT y + cT z and aT x + bT y + cT z . Here are some consequences of our main theorems: (i) All sufficiently large odd numbers have the form 2αx 2 + y 2 + z 2 if and only if all prime divisors of a are congruent to 1 modulo 4. (ii) The form αx 2 + y 2 + T z is almost universal (i.e., it represents sufficiently large integers) if and only if each odd prime divisor of a is congruent to 1 or 3 modulo 8. (iii) αx 2 +T y + T z is almost universal if and only if all odd prime divisors of a are congruent to 1 modulo 4. (iv) When υ 2 (α) ≠ 3, the form αT x + T y + Tz is almost universal if and only if all odd prime divisors of α are congruent to 1 modulo 4 and υ 2 (α) ≠ 5, 7, ..., where υ 2 (a) is the 2-adic order of α.