On two Romanoff type problems of Erd\H{o}s

Abstract

Let $\mathcal{P}$ be the set of primes and, for $a>1$, put $\mathcal{S}_a={p+\lfloor a^k\rfloor:p\in\mathcal{P},\ k\ge1}$. Erd\H{o}s recorded a question of Kalm'ar asking whether $\mathcal{S}_a$ has positive lower asymptotic density for every real $a>1$. We prove this for almost every $a>1$, with $$ \liminf_{N\to\infty}\frac{|\mathcal{S}_a\cap[1,N]|}{N} \ge \frac{1}{\log a+9C_0/\pi^2}, $$ where $C_0$ is an absolute constant. The dependence on $a$ has the correct order $1/\log a$ as $a\to\infty$. For almost every $a>1$ and every $\eta>0$, we also prove that at least $x^{1-\eta}$ positive integers $n\le x$ lie outside $\mathcal{S}_a$ for all sufficiently large $x$. For the golden ratio $\varphi=(1+\sqrt5)/2$, the corresponding sumset has positive lower asymptotic density and upper asymptotic density at most $1937/1938$. We also consider a problem of Erd\H{o}s asking whether every sufficiently large odd integer is the sum of a squarefree integer and a power of two. Replacing $2^m$ by $\lfloor a^m\rfloor$, we prove for almost every $a>1$ that the exceptional set up to $x$ is $$ O_{a,\varepsilon}\!\left(\frac{x(\log\log x)^{1+\varepsilon}}{\sqrt{\log x}}\right), $$ and, allowing two distinct exponents, it is $$ O_{a,\varepsilon}\!\left(\frac{x(\log\log x)^{1+\varepsilon}}{\log x}\right). $$ The proofs use metric residue distribution and weighted pair correlation estimates for $\lfloor a^k\rfloor$, together with a congruence covering argument for the prime exceptional set.