Let $b,c,d\in\mathbb N=\{0,1,2,\ldots\}$ with $1\le b\le c\le d$. For a subset $S$ of $\mathbb N=\{0,1,2,\ldots\}$, if $$\{w^2+bx^2+cy^2+dz^2:\ w,x,y,z\in S\}=\mathbb N$$ then we say that $w^2+bx^2+cy^2+dz^2$ is universal over $S$, or $S$ is a $(b,c,d)$-universal set.
Lagrange's four-square theorem indicates that $\mathbb N$ is a $(1,1,1)$-universal set. As observed by S. Ramanujan and proved by L. E. Dickson in 1927, for positive integers $b\le c\le d$, $\mathbb N$ is $(b,c,d)$-universal if and only if $(b,c,d)$ is among the following 54 triples: \begin{align*}&(1,1,1),\,(1,1, 2),\,(1,2, 2),\, (2,2, 2),\, (1,1, 3),\, (1,2, 3), \\&(2,2, 3),\,(1,3, 3),\,(2,3, 3),\,(1,1, 4),\,(1,2, 4),\,(2,2, 4), \\&(1,3, 4),\,(2,3, 4),\,(2,4, 4),\,(1,1, 5),\,(1,2, 5),\,(2,2, 5), \\&(1,3, 5),\,(2,3, 5),\,(2,4, 5),\,(1,1, 6),\,(1,2, 6),\,(2,2, 6), \\&(1,3, 6),\,(2,3, 6),\,(2,4, 6),\,(2,5, 6),\,(1,1, 7),\,(1,2, 7), \\&(2,2, 7),\,(2,3, 7),\,(2,4, 7),\,(2,5, 7),\,(1,2, 8),\,(2,3, 8), \\&(2,4, 8),\,(2,5, 8),\,(1,2, 9),\,(2,3, 9),\,(2,4, 9),\,(1,5, 9), \\&(1,2,10),\,(2,3,10),\,(2,4,10),\,(2,5,10),(1,2,11),(2,4,11), \\&(1,2,12),\,(2,4,12),\,(1,2,13),\,(2,4,13),\,(1,2,14),\,(2,4,14). \end{align*}
We call a subset $S$ of $\mathbb N$ sparse if $$\lim_{x\to+\infty}\frac{|\{n\le x:\ n\in S\}|}x=0.$$
QUESTION. Whether for some positive integers $b\le c\le d$ there is a sparse subset $S$ of $\mathbb N$ such that $S$ is $(b,c,d)$-universal?
By the Prime Number Theorem, the set of all primes is a sparse subset of $\mathbb N$. Recall that a postive integer $n$ is said to be practical if each $m=1,\ldots,n$ can be written as a sum of distinct (positive) divisors of $n$. For the list of practical numbers, see https://oeis.org/A005153. It is well known that $1$ is the only odd practical numbers. The set of all practical numbers is also known as a sparse subset of $\mathbb N$.
Motivated by the above question, I consider two sparse sets $$S_1=\left\{\frac{p-1}2:\ p\ \text{is}\ 1\ \text{or an odd prime}\right\}$$ and $$S_2=\{m\in\mathbb N:\ 2m\ \text{is zero or a practical number}\},$$ and check numerically whether $S_1$ or $S_2$ is $(b,c,d)$-universal for some $(b,c,d)$ among the 54 listed triples. As a result of this inverstigation, I have formulated the following conjecture.
Conjecture. Let $b,c,d$ be positive integers with $b\le c\le d$.
(i) $S_1$ is $(b,c,d)$-universal if and only if $(b,c,d)$ is among the following $25$ triples: \begin{align*}&(1,2,3),\,(1,2,4),\,(1,2,5),\,(1,2,6),\,(1,2,7),\,(1,2,12),\,(1,3,5),\,(1,3,6), \\&(2,2,3),\,(2,2,5),\,(2,3,3),\,(2,3,4),\,(2,3,5),\,(2,3,7),\,(2,3,9),\,(2,3,10), \\&(2,4,4),\,(2,4,5),\,(2,4,9),\,(2,4,10),\,(2,4,11),\,(2,4,12),\,(2,5,7),\,(2,5,8),\,(2,5,9). \end{align*}
(ii) $S_2$ is $(b,c,d)$-universal if and only if $(b,c,d)$ is among the following $20$ triples: \begin{align*}&(1,2,2),\,(1,2,3),\,(1,2,5),\,(1,2,6),\,(1,2,7),\,(1,2,8),\,(1,2,10), \\&(1,2,11),\,(1,2,14),\,(2,2,3),\,(2,2,5),\,(2,2,7),\,(2,3,3),\,(2,3,5),\\&(2,3,6),\,(2,3,7),\,(2,3,10),\,(2,4,5), \,(2,5,6),\,(2,5,8). \end{align*}
For example, the $(2,4,4)$-universality of $S_1$ says that each $n\in\mathbb N$ can be written as $$(p-1)^2+(q-1)^2+\frac{(r-1)^2}2+\frac{(s-1)^2}4,$$ where each of $p,q,r,s$ is either $1$ or an odd prime. The $(1,2,2)$-universality of $S_2$ (cf. https://oeis.org/A389523) says that each $n\in\mathbb N$ can be written as $w^2+x^2+2y^2+2z^2$ with $w,x,y,z\in S_2$, which is stronger than Lagrange's four-square theorem since $2y^2+2z^2=(y+z)^2+(y-z)^2$. The "only if" part of the conjecture is easy.
Any comments on the question or the above related conjecture are welcome!
