Konieczny 2016 sets recurrence as bases positive integers

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lemma_1_1: Konieczny's obstruction for constant thresholds: if N is odd and N alpha is within (1 - delta)/(kN) of m/k modulo 1, with k even and m odd, then N is not a sum of two elements of the set of n with alpha n^2 within eps(n) of an integer whenever eps(n) <= delta/(2k).

lemma_1_2: Konieczny's obstruction for thresholds tending to 0: an increasing sequence of odd N_i with N_i alpha = m_i/k + gamma_i/(k N_i), k even, m_i odd, and gamma_i accumulating at some gamma with |gamma| < 1, has N_i outside 2A for infinitely many i, unless gamma + k n^2 alpha is an integer for some integer n.

proposition_1_3: Konieczny's sqrt2 case: the set of n with sqrt2 n^2 within eps(n) of an integer is not a basis of order 2 when eps(n) tends to 0 or is bounded by a constant below (1 - 1/(4 sqrt2))/4, and in the bounded case at least a constant times log T integers up to T are missed.

proposition_2_9: Konieczny's exceptional values: if the continued fraction of alpha has convergent denominators and partial quotients with prescribed growing divisibility, and log a_i / i tends to 0, then the set of n with alpha n^2 within a constant eps_0 > 0 of an integer is a basis of order 2; uncountably many such alpha exist.

question_1: Erdős's question, recorded as the paper's Question 1, whether the set of n with sqrt2 n^2 within 1/log n of an integer is a basis of order 2; the paper says the answer is negative and proves it in Proposition 1.3.

question_2: The paper's Question 2, the weaker almost-basis form of Question 1, which it answers positively: A + A has asymptotic density 1, and the introduction states that the complement of A + A up to T is O(log^C T).

theorem_2_6: Konieczny's almost-basis theorem: for irrational alpha the sumset of the set of n with alpha n^2 within eps(n) of an integer has density 1 once eps is above an alpha-dependent rate; for finite irrationality measure the rate can be any eps(n) = n^(-o(1)) and the complement up to T is O(T^(1-c)), and for badly approximable alpha and constant eps it is O(log T).

theorem_3_1: Konieczny's general higher-degree theorem: for an affine family P of real polynomials and eps(n) = n^(-o(1)), either every member has degree at most 2, or some member's degree exceeds that of its difference with every member, or the recurrence set of almost every p in P is a basis of order 2.

theorem_a: Konieczny's Theorem A on A = {n : ||alpha n^2|| <= eps(n)} with alpha irrational: for eps decaying slowly enough A is an almost basis of order 2 (A1), a basis of order 2 for uncountably many alpha (A2) and a basis of order 3 for every alpha (A3); for almost all alpha it is not a basis of order 2 whenever eps(n) tends to 0 (A4).

theorem_a4: Konieczny's precise form of A4: outside a Lebesgue-null set of alpha, and for every irrational alpha in Q(sqrt d), the set of n with alpha n^2 within eps(n) of an integer is not a basis of order 2 for any eps(n) tending to 0, with companion results for small constant eps.

theorem_b: Konieczny's degree-three-and-higher result: for d >= 3 and eps(n) = n^(-o(1)), the set of n with alpha n^d within eps(n) of an integer is a basis of order 2 for almost all alpha (B1), while for an uncountable closed set of alpha and every eps below a small constant it is not (B2).

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Konieczny, Jakub, Sets of recurrence as bases for the positive integers. Acta Arith. 174 (2016), no. 4, 309--338, doi:10.4064/aa8125-4-2016. The copy read for this card is the arXiv preprint arXiv:1504.02410v3 (19 Jul 2018), whose page numbers are cited here. The arXiv record names arXiv's non-exclusive distribution license, every other right reserved.

For a real polynomial $p$ and slowly decaying $\epsilon$, the paper asks when the recurrence set $\{n\in\mathbb N:\|p(n)\|_{\mathbb R/\mathbb Z}\le\epsilon(n)\}$ is a basis of finite order for $\mathbb N$, that is when $k\mathcal A$ contains all large integers, and when it is only an almost basis, meaning $k\mathcal A$ has asymptotic density $1$. In degree one the paper says (p. 2) that for irrational $\alpha$ the set $\{n:\|\alpha n\|\le\epsilon(n)\}$ is not a basis of order $k$ when $\epsilon(n)<1/(3k)$ for all $n$, or when $\epsilon(n)\to0$, leaving the details to the reader. Degree two is the interesting case. Question 1 (p. 2) records Erdős's question, which the paper attributes to a personal communication from Ben Green, whether $\{n:\|\sqrt2\,n^2\|\le1/\log n\}$ is a basis of order $2$; the answer is negative (Proposition 1.3, p. 6). The introduction's formula for the integers missed lacks a factor $1/2$: the proof takes $N_i=b_i/2$ for odd $i$, where $(3+2\sqrt2)^i=a_i+b_i\sqrt2$, while the printed value is the even integer $b_{2i+1}$. Question 2 (p. 2), the almost-basis form, is answered yes, and the introduction states the bound $|[T]\setminus2\mathcal A|\ll\log^CT$, which no numbered statement prints for this set; Theorem 2.6 gives $T^{1-c}$.

Theorem A (p. 2) collects the degree-two picture for $\{n:\|\alpha n^2\|\le\epsilon(n)\}$, $\alpha$ irrational: for $\epsilon$ above an $\alpha$-dependent rate tending to $0$, the set is an almost basis of order $2$ (A1), a basis of order $2$ for uncountably many exceptional $\alpha$ (A2), and a basis of order $3$ for every irrational $\alpha$ (A3); for almost all $\alpha$ it is not a basis of order $2$ whenever $\epsilon(n)\to0$ (A4), and Section 1 proves this also for every irrational $\alpha$ in a real quadratic field. The precise forms in the body use strict inequality, $\|\alpha n^2\|<\epsilon(n)$ ((1.1), p. 4). Theorem B (p. 3, precise forms p. 20) shows that in degree $d\ge3$ the set is a basis of order $2$ for almost all $\alpha$ when $\epsilon(n)=n^{-o(1)}$, from the general Theorem 3.1 on affine families of polynomials, but not for the $\alpha$ of a closed uncountable set whenever $\epsilon(n)\le\epsilon_0$ for a small constant $\epsilon_0$.

Source: https://arxiv.org/abs/1504.02410.

Read status: claims checked for every statement linked below, read clause by clause on the page images of the print; the proofs of Lemmas 1.1 and 1.2 and Proposition 1.3 were followed, the others read in outline. Nothing here is independently reviewed.

Bears on. #1147: the problem asks whether $\{n\ge1:\|\alpha n^2\|<1/\log n\}$ is a basis of order $2$ for irrational $\alpha>0$. Proposition 1.3 (p. 6) says it is not for $\alpha=\sqrt2$, the paper's negative answer to Question 1, and Theorem (A4 reiterated) (p. 4) says it is not for $\alpha$ outside a Lebesgue-null set or in $\mathbb Q[\sqrt d]\setminus\mathbb Q$, since $1/\log n\to0$. A2 (Proposition 2.9) gives bases of order $2$ for uncountably many $\alpha$ only above a rate the paper does not compare with $1/\log n$, so the paper decides nothing for the remaining $\alpha$.

Results.

• Question 1 (p. 2): is $\{n:\|\sqrt2\,n^2\|\le1/\log n\}$ a basis of order $2$? Answered no.
• Question 2 (p. 2): is the same set an almost basis of order $2$? Answered yes.
• Theorem A (pp. 2--3): items A1 to A4 for $\{n:\|\alpha n^2\|\le\epsilon(n)\}$.
• Theorem B (p. 3, reiterated p. 20): items B1 and B2 for $\alpha n^d$, $d\ge3$.
• Theorem (A4 reiterated) (p. 4), with Propositions 1.4 (p. 6), 1.5 (p. 8) and 1.6 (p. 10): not a basis of order $2$ for almost all $\alpha$ and for quadratic irrationals when $\epsilon(n)\to0$.
• Lemma 1.1 (p. 4): the obstruction to $N\in2\mathcal A_\epsilon^\alpha$ for constant $\epsilon$.
• Lemma 1.2 (p. 5): the obstruction for $\epsilon(n)\to0$.
• Proposition 1.3 (p. 6): the case $\alpha=\sqrt2$.
• Theorem 2.6 (pp. 15--16): almost bases of order $2$, with complement bounds.
• Proposition 2.9 (p. 18), with Observation 2.8 and Theorem (A2 reiterated) (p. 12): bases of order $2$ for exceptional $\alpha$.
• Theorem 3.1 (p. 20): affine families of polynomials in higher degree.

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