On monochromatic sums of squares and primes
For a subset \({\mathcal A}\subseteq {\mathcal U}\), one typical inverse additive question is whether any element in \(\mathcal U\) is a sum of elements in \(\mathcal A\), and \(\mathcal A\) is called an additive basis of \(\mathcal U\) if so. \(\mathcal A\) is called the additive basis of order \(h\) if any element of \(\mathcal U\) is a sum of at most \(h\) elements of \(\mathcal A\). In the case of \({\mathcal U}={\mathbb Z}\), \(\mathcal A\) is called an asymptotic basis if any large enough integer is a sum of elements in \(\mathcal A\). The asymptotic order can be defined in a similar way. Another variation is to add the coloring to the problem. A coloring of \(\mathcal A\) is a partition of \(\mathcal A\). A monochromatic sum is a sum taken from one of the sets in the given partition. For any positive integer \(K\), we can define the \(K\)-chromatic order ord\(_K\) as the maximum order of monochromatic sum over any \(K\)-partition. This paper looks at two special sets: \({\mathcal A}={\mathcal Q}\), the set of squares, and \({\mathcal A}={\mathcal P}\), the set of primes. For \({\mathcal A}={\mathcal Q}\), the paper gives the following upper and lower bounds \[ K\exp\left((\log2+o(1))\frac{\log K}{\log\log K}\right)\leq\text{ ord}_K(\mathcal Q)\leq c_3(K\log K)^5 \] where \(c_3\) can be taken equal to \(10^9\). For \({\mathcal A}={\mathcal P}\), the following upper and lower bounds are given \[ (e^\gamma+o(1))K\log\log K\leq\text{ ord}_K(\mathcal Q)\leq 1500K^3. \] In the case of a hybrid scenario, \({\mathcal A}=\{1\}\cup\{hq\,| \,q\geq0\}\), not every basis has a \(K\)-chromatic order, which leads to the definition of \textit{well sifted}. For any prime \(p\), denote \({\mathcal A}^{(p)}=\{a\in{\mathcal A}\,| \,p| a\,\}\). For any real number \(\tau<1\), \(\mathcal A\) is well \(\tau\)-sifted if there exists a real number \(\tau>0\) such that \[ \mathop{\lim\sup}_{N\to\infty}\frac{A^{(q_1,q_2,\dots,q_K)}(N)}{A(N)}\leq1-\tau \] For a well \(\tau\)-sifted basis, similar results are adopted.
scientific article; zbMATH DE number 5151586