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Claim. For real $y>1$ let $S_y={p+\lfloor y^k\rfloor:p \text{ prime}, k\ge1}$ and let $\delta_y$ be its lower density. Theorem 1.1 of Yuchen Ding's paper (card), present since its first version of 17 March 2025 under the title On a Romanoff type problem of Erdős and Kalmár, states that for Lebesgue-almost every real $y>1$

$$\delta_y\ge\frac{1}{\log y+9C_0/\pi^2},$$

with $C_0$ an absolute constant defined in the paper; the paper notes that the order $1/\log y$ is best possible as $y\to\infty$, by the prime number theorem. Theorem 1.2 of the fifth version (10 September 2026, retitled On two Romanoff type problems of Erdős) states that for the golden ratio $\varphi=(1+\sqrt5)/2$

$$0<\liminf_{x\to\infty}\frac{\lvert S_\varphi\cap[1,x]\rvert}{x} \le\limsup_{x\to\infty}\frac{\lvert S_\varphi\cap[1,x]\rvert}{x} \le\frac{1937}{1938},$$

with at least $x/1938-O(\log x)$ integers $n\le x$ outside $S_\varphi$. The lower bound comes from the Romanoff-type theorem of Ballot and Luca for linear recurrences (Acta Arith. 161 (2013), 33--46, Theorem 1), applied to the odd-indexed Lucas numbers $L_{2j+1}=\lfloor\varphi^{2j+1}\rfloor$; the upper bound comes from a finite covering congruence for the Lucas numbers. The paper remarks that starting $k$ at $1$ rather than $0$ does not change any lower density, since the omitted integers $p+1$ have density zero. The method of Theorem 1.1 combines Romanoff's second-moment argument with the uniform distribution of $y^k$ modulo $1$ for almost every $y$ (Koksma's theorem) and exponential-sum estimates for the distribution of $\lfloor y^k\rfloor$ in residue classes.

Covers. $C=\varphi$, answered yes. Theorem 1.1 decides no specified $C$: it gives a yes for almost every $C>1$ without naming any. Not covered: every other non-integer $C>1$; the integers are Romanoff's claim.

Depends on. Nothing on this wiki; the claim rests on the cited preprint.

Standing. The paper is on arXiv only, and its record carries no journal reference, so the claim is claimed with no evidence listed. The site's commentary credits the almost-all theorem on a problem it labels OPEN, which is not acceptance; the golden-ratio case first appears in the fifth version, and the site does not record it. The page is named by the first version's posting date.