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Statement

Question 1 (p. 2). With $\|t\|_{\mathbb R/\mathbb Z}$ the distance from $t$ to the nearest integer and $\mathbb N=\{0,1,2,\ldots\}$, is the set

$$\mathcal A=\Bigl\{n\in\mathbb N:\ \bigl\|\sqrt2\,n^2\bigr\|_{\mathbb R/\mathbb Z}\le\frac1{\log n}\Bigr\}$$

a basis of order $2$, that is, does $\mathcal A+\mathcal A$ contain every sufficiently large integer?

The paper attributes the question to Erdős; its footnote (p. 2) gives the source as a personal communication from Ben Green and says no written reference could be located.

The paper's answer (p. 2). The answer is negative. The introduction says one can produce an explicit sequence of integers $N_i\notin2\mathcal A$ for all sufficiently large $i$, and prints it as "$N_i=\frac{(3+2\sqrt2)^{2i+1}-(3-2\sqrt2)^{2i+1}}{2\sqrt2}$" [sic]. Writing $(3+2\sqrt2)^j=a_j+b_j\sqrt2$, that number is $b_{2i+1}$, which is even. The proof of Proposition 1.3 (p. 6) instead takes $N_j=b_j/2$ for odd $j$, which is odd; the printed formula lacks a factor $1/2$. The paper adds that several other constructions of this type exist, each giving a sequence outside $2\mathcal A$ growing exponentially.

Proposition 1.3 concerns the set with strict inequality $\|\sqrt2\,n^2\|_{\mathbb R/\mathbb Z}<\epsilon(n)$, for any $\epsilon(n)\to0$ (definition (1.1), p. 4); Question 1 is printed with $\le$.

Proof pointer

The negative answer is Proposition 1.3 (p. 6), through Lemma 1.2 (p. 5).

Read depth

Claims checked: Question 1, its footnote and the paragraph after it on p. 2 were read clause by clause on the page images of the print, and the formula was compared with (1.4) and (1.5) on p. 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

• Problem 1147: Question 1 is the problem's question for the single value $\alpha=\sqrt2$, printed with $\le1/\log n$ where the problem writes $<1/\log n$, and the paper answers it no. The problem asks it for an irrational $\alpha>0$; the paper's results for other $\alpha$ are on the Theorem A4 page.