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Updated Oct 7

Problem 1192

claims/: The 1 claim page of Problem 1192, one per claimant's result; the problem's standing derives from them.

Statement. For $A\subset \mathbb{N}$ let $f_r(n)$ count the number of solutions to $n=a_1+\cdots+a_r$ with $a_i\in A$ .

Does there exist, for all $r\geq 2$ , a basis $A$ of order $r$ (so that $f_r(n)>0$ for all large $n$ ) such that

\sum_{n\leq x}f_r(n)^2 \ll x

for all $x$ ?

Status. Open. Ruzsa [Ru90] answers the case $r=2$ , the accepted partial claim Ruzsa; the cases $r\ge3$ are open.

Source. erdosproblems.com/1192, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1192, https://www.erdosproblems.com/1192.

References.

[Ru90] Ruzsa, Imre Z., A just basis. Monatsh. Math. (1990), 145-151.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

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Linked library material

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Linked from (21)

Ruzsa's just basis of order two problems/additive_bases/E1192/claims Additive Bases and Sidon Sets additive_bases/ding_2026_improved_upper_bound_ruzsa_number Lemma 2.2 (p. 3): the Ruzsa number R_m is at most 116 for m up to 132^2 Theorem 1.1 (p. 1): the Ruzsa number R_m is at most 128 for every modulus m Theorem 1.3 (p. 2): the Ruzsa number of 8p^2 is at most 32 for every prime p additive_bases/erdos_1960_additive_properties_random_sequences_positive_integers additive_bases/erdos_1990_representations_integers_as_sum_k_terms Theorem 1 (p. 246): for every fixed k an asymptotic basis of order k with r_k(n) of order log n Theorem 2 (p. 254): almost always r_k(n) < [6 b_2 c k + o(1)] log n for large n Theorem 3 (p. 258): almost always r_k(n) > C_1 log n for large n additive_bases/green_2026_100_open_problems additive_bases/konyagin_2009_erdos_turan_problem_infinite_groups Corollary 1: bases with representation function bounded by an absolute constant Theorem 1: perfect bases of order two in infinite abelian groups with |2G| = |G|Theorem 2: in exponent 2, a basis with at most 36 representations of each non-zero element additive_bases/nathanson_2014_paul_erdos_additive_bases additive_bases/ruzsa_1990_just_basis Theorem 1: a set in [0, 3p^2] of size at most 12p whose sumset covers [2p^2, 4p^2] with at most 288 representations Theorem 2: a basis of order 2 whose representation counts are bounded in square mean

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