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Statement

Question 2 (p. 2). Is the set $\mathcal A=\{n\in\mathbb N:\ \|\sqrt2\,n^2\|_{\mathbb R/\mathbb Z}\le1/\log n\}$ of Question 1 an almost basis of order $2$, that is, does $2\mathcal A=\mathcal A+\mathcal A$ have asymptotic density $1$? Asymptotic density is $d(\mathcal B)=\lim_{n\to\infty}|\mathcal B\cap[n]|/n$ when the limit exists, with $[n]=\{1,\ldots,n\}$ (p. 1).

The paper's answer (p. 2). Yes. The introduction states the stronger bound

$$\bigl|[T]\setminus2\mathcal A\bigr|\ll\log^CT\qquad(T\to\infty),$$

where $C$ is a constant.

The general result proved in the paper is Theorem 2.6 (pp. 15--16), stated for the sets of (1.1), defined with strict inequality. As printed, it gives the bound $T^{1-c}$ for $\alpha$ of finite irrationality measure when $\log(1/\epsilon(n))/\log n\to0$, and the bound $\log T$ only for badly approximable $\alpha$ with $\epsilon(n)\ge\epsilon_0>0$ for all $n$. No numbered statement of the paper prints the bound $\log^CT$ for $\epsilon(n)=1/\log n$.

Proof pointer

Theorem 2.6 and its proof (pp. 16--17).

Read depth

Claims checked: Question 2 and the paragraph after it on p. 2 were read clause by clause on the page images of the print and compared with the statement of Theorem 2.6. Nothing here is independently reviewed.

Dependencies

None in the corpus.

Source. J. Konieczny, Sets of recurrence as bases for the positive integers, Acta Arith. 174 (2016), no. 4, 309--338, doi:10.4064/aa8125-4-2016; the edition read and its page numbers are named on the source card.

Bears on

None directly: Problem 1147 asks for a basis of order $2$, not an almost basis.