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Statement

Setting (p. 1). $N_f^+(x)$ is the set of $n\le x$ with $n=k+f(k)$ for some $k$, and in the bounds it stands for its size.

Theorem 1.3 (p. 2). Let $f\colon\mathbb N\to\mathbb Z$ satisfy $0\le f(k)\le ck$ for some $c>0$. Then

$$x-N_f^+(x)=\#\{n\le x:n\ne k+f(k)\}\ \ge\ \frac1{(2c+2)x}\sum_{k\le x}f(k).$$

The paper remarks (p. 2) that the bound is tight up to the constant $(2c+2)^{-1}$ in general, by the examples $f\equiv1$ and $f(k)=k$, and not tight at all for $f$ equal to $1$ on odd $k$ and $0$ on even $k$.

Application to the totient (p. 2). With $c=1$ and $\sum_{k\le x}\varphi(k)=(3/\pi^2+o(1))x^2$, the theorem gives, for large $x$,

$$N_\varphi^+(x)\le\Bigl(1-\frac3{4\pi^2}+o(1)\Bigr)x\le0.93x ,$$

the upper bound in item (3) of the abstract. The print introduces this as an application of "Theorem 1.2" [sic]; the bound is the case $f=\varphi$ of Theorem 1.3.

Proof pointer

Section 4, p. 10, after an idea of Zannier. Let $A_s(x)$ be the set of $n\le x$ with exactly $s$ representations $n=k+f(k)$. Since every representation of an $n\le x$ uses some $k\le x$, counting gives $\sum_{s\ge2}(s-1)|A_s(x)|\le|A_0(x)|$, so the set $B_1(x)$ of $k\le x$ whose value $k+f(k)$ is represented only once satisfies $x-|B_1(x)|\le2|A_0(x)|$. Comparing $\sum_{k\le x}(k+f(k))$ with the sum of the integers $n\le x$ uniquely represented gives $\sum_{k\le x}f(k)\le(c+1)x\,(x-|B_1(x)|)$, and the theorem follows.

Read depth

Claims checked: the statement, the remarks and the totient application were read on the print (arXiv v1, p. 2), and the proof on p. 10 was followed. Nothing here is independently reviewed.

Dependencies

None in the corpus. External input: the mean value of $\varphi$, for the application.

Source. M. R. Gabdullin, V. V. Iudelevich and F. Luca, Numbers of the form $k+f(k)$, J. Number Theory 262 (2024), 58--85, doi:10.1016/j.jnt.2024.03.010; arXiv:2306.16035. Labels and pages are those of the arXiv v1 edition named on the source card.

Bears on

• Problem 822: the application bounds the upper density of the integers of the form $n+\varphi(n)$ by $0.93$. The problem asks about their lower density, which this bound does not decide; the lower bound is Theorem 1.4.