Turturean 2026 negative answer erdos problem 870

../

lemma_5_1: States that for integers h at least 2 and L at least 1 there are positive integers N < M such that F = [1,N] gives at least L nondecreasing h-tuples with each sum modulo M, and some residue tau modulo M is attained by sums of at most h+1 fillers only as tau itself, using at least h fillers.

proposition_2_3: States that there are an absolute constant eta_2 > 0 and a set A of positive integers with A together with A+A cofinite, r_A(n) at least eta_2 log n for all large n, A(x) = o(x), and no minimal additive basis of order 2 in the at-most-two sense contained in A.

proposition_3_4: States that there is an absolute constant eta_3 > 0 such that for every finite list of pairs (U,V) of finite sets of nonnegative integers with U nonempty and every finite P_0 there is a set A, disjoint from P_0, with A+A cofinite, r_A(n) at least eta_3 log n, density zero, and a deletion property for the sets Phi_{U,V}(D).

proposition_4_1: States that for every C > 0 there is a set E of positive integers that is an additive basis of order 3, has R_{E,3}(n) at least C log n for all large n, and contains no minimal additive basis of order 3.

proposition_5_2: States that for every integer k at least 4 and every C > 0 there is a set E of positive integers that is an additive basis of order k, has R_{E,k}(n) at least C log n for all large n, and contains no minimal additive basis of order k.

theorem_1_1: States that for every integer k at least 3 and every real C > 0 there is a set E of positive integers that is an additive basis of order k, has R_{E,k}(n) at least C log n for all large n, and contains no minimal additive basis of order k.

---

David Turturean, A Negative Answer to Erdős Problem #870. Overleaf preprint (2026). The edition read is dated April 2026 on its first page and runs to 11 pages; labels and pages below are those of that edition.

Theorem 1.1 (p. 2) is the negation of Erdős Problem #870 in the problem site's wording: for every integer k>=3 and every real C>0 there is a set E that is an additive basis of order k, every large integer being a sum of at most k elements, has R_{E,k}(n) >= C log n for all large n, where R_{E,k}(n) counts nondecreasing representations by at most k elements, and contains no minimal order-k basis. The input is the random order-2 basis of Larsen and Larsen (arXiv:2601.18507), built in stages I_n=[2^{2^n},2^{2^{n+1}}) by Bernoulli sampling, deleting the summands of a sparse set of 'canary' elements and adding restoration elements; Section 2 repairs it for the at-most-two convention with Lemma 2.1 (A(x)=o(x) for all x) and Lemma 2.2 (exclusion of canary representations through an old summand), packaged as Proposition 2.3. For k>=4, Section 5 uses a finite-filler reduction (Lemma 5.1, Proposition 5.2): E = M·A ∪ ([1,N]∩N) with M > N, so that a rigid residue class mod M makes any order-k subbasis descend, modulo M, to an at-most-two subbasis of A. For k=3, Section 3 replaces each canary by a cluster of finitely many shifts of a random center and excludes accidental representations across clusters by a summable Borel-Cantelli bound (Lemmas 3.1-3.3, Proposition 3.4), and Proposition 4.1 (Section 4) takes E = 2A ∪ F with a finite set F of even and odd fillers. Section 6 (p. 11) assembles Theorem 1.1 from Propositions 4.1 and 5.2. The probabilistic core cites Larsen-Larsen internals (their Lemmas 2, 6 and 7, Proposition 5 and finite-incidence argument) rather than reproving them. The acknowledgments (p. 11) say the construction and proof were produced by an automated scaffold designed by the author that queried GPT-5.4 Pro and then GPT-5.5 Pro, and that the author independently verified the final proof. The claim is recorded on its claim page, Turturean.

Source: https://www.overleaf.com/read/gknkvvxrymfv. No notice is printed on the file's first or last pages; the hosting site's terms speak for the site, not the paper: Overleaf's terms page (https://www.overleaf.com/legal, read 2026-10-02) states "We don't claim any ownership of your stuff" and grants readers of a shared project no license; the term is unstated.

Results

• Theorem 1.1 (p. 2): for every integer $k\ge3$ and every real $C>0$, a set $E\subseteq\mathbb N$ that is an additive basis of order $k$, has $R_{E,k}(n)\ge C\log n$ for all sufficiently large $n$, and contains no minimal additive basis of order $k$.
• Proposition 2.3 (p. 3), with Lemmas 2.1 (p. 2) and 2.2 (p. 3): an absolute $\eta_2>0$ and a set $A$ with $A\cup(A+A)$ cofinite, $r_A(n)\ge\eta_2\log n$ for all large $n$, $A(x)=o(x)$, and no minimal additive basis of order 2 in the at-most-two sense inside $A$.
• Proposition 3.4 (pp. 5–6): the clustered order-2 input, a set $A$ avoiding a given finite $P_0$, with $A+A$ cofinite, $r_A(n)\ge\eta_3\log n$, $A(x)=o(x)$, and an element of any $D\subseteq A$ with $\Phi_{U,V}(D)$ cofinite whose deletion keeps $\Phi_{U,V}$ cofinite and leaves an order-3 basis.
• Proposition 4.1 (p. 8): the case $k=3$ of Theorem 1.1.
• Lemma 5.1 (p. 9): the finite filler set $[1,N]$ with $M>N$, every residue mod $M$ hit by at least $L$ nondecreasing $h$-tuples, and a rigid residue $\tau$.
• Proposition 5.2 (p. 9): the cases $k\ge4$ of Theorem 1.1.

Read status. Claims checked for the results above, read clause by clause on the print; the proofs were followed, and the Larsen-Larsen results the paper cites were not read.

Bears on. #870: Theorem 1.1 asserts, for every $k\ge3$, an order-$k$ basis in the at-most-$k$ sense with at least $C\log n$ nondecreasing representations of each large $n$ by at most $k$ elements, for arbitrary $C$, and no minimal order-$k$ subbasis; the paper calls this the negation of the problem's threshold assertion. It does not treat the version of the problem page's source, with representations by exactly $h$ elements counted as disjoint representations.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.