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Halász proves, as Theorem 2 of his 1977 paper, that for $n$ vectors $\mathbf a_1,\ldots,\mathbf a_n\in\mathbb R^d$ with $|\mathbf a_k-\mathbf a_{k'}|\ge1$ for $k\ne k'$ such that, for some $\delta>0$ and every unit vector $\mathbf e$, at least $\delta n$ of them satisfy $|(\mathbf a_k,\mathbf e)|\ge1$, at most $c(\delta,d)2^nn^{-1-d/2}$ of the $2^n$ signed sums $\sum_k\varepsilon_k\mathbf a_k$ ($\varepsilon_k=\pm1$) lie in any open unit ball. His remark after the theorem records that this confirms a conjecture Erdős had communicated to him: with $\mathbf a_k=(a_k,1)$ and $|a_k-a_{k'}|\ge1$ the count is at most $c2^nn^{-2}$, which is the Sárközy--Szemerédi problem with the number of plus signs also fixed, and he names this question as the starting point of his investigations in higher dimensions.

For the second question of Problem 362, with $A=\{a_1<\cdots<a_N\}\subseteq\mathbb N$: the subsets $S\subseteq A$ with $|S|=l$ and $\sum S=t$ are the sign vectors whose sum $\sum_k\varepsilon_k(a_k,1)$ equals $(2t-\sum A,\,2l-N)$, so their number is at most the number of signed sums in the unit ball around that point. The vectors $(a_k,1)$ are $1$-separated, and the fixed-size count does not change when $A$ is translated, so after moving the middle element of $A$ to $0$ at least $N/4$ of the vectors satisfy the condition of the theorem for $N\ge6$; the count is therefore at most $c(1/4,2)2^N/N^2$, with $N\le5$ covered by the trivial bound. That reduction is the problem page's authored line, detailed on the library's result page, and is not spelled out in Halász's remark.

The paper's library card is Halász 1977, with a result page for Theorem 2. The statements and the remark are checked against the paper; the one-paragraph proof of Theorem 2, a modification of the proof of his concentration inequality (Theorem 4), was followed and not checked, and nothing here is independently reviewed.

Covers. The second question of Problem 362: for every $N$-element $A\subseteq\mathbb N$, every $l$ and every $t$, the number of subsets of $A$ of size $l$ with sum $t$ is $\ll2^N/N^2$ with an absolute constant. The first question has Sárközy and Szemerédi's page.

Formalization. The statement file of the formal-conjectures project (FormalConjectures/ErdosProblems/362.lean) states the second question as its declaration erdos_362.variants.fixed_card, marks it solved and points, through its formal_proof attribute, at a Lean 4 file in Boris Alexeev's lean-proofs repository, linked above at the commit the attribute pins. That file declares itself a formalization of a solution to Problem 362, names András Sárközy, Endre Szemerédi and Gábor Halász as its informal authors and Codex and GPT-5.6 Sol as its formal authors, and cites this paper for the fixed-cardinality estimate. The second conjunct of its theorem erdos_362, a bound $C\,2^N/N^2$ on the number of subsets of $A$ of size $l$ with sum $t$ (fixedCardSubsetSumFiber A l t) over all nonempty finite $A\subseteq\mathbb N$ and all $l$ and $t$, is this claim; the first conjunct is the claim of Sárközy and Szemerédi's page. The file contains no sorry, no axiom command and no native_decide at the pinned commit. This project has not built the file or audited its statement against the problem, so it supplies no formalized evidence; the site's label PROVED (LEAN) refers to this development.

Acceptance. The paper is refereed: G. Halász, Estimates for the concentration function of combinatorial number theory and probability, Period. Math. Hungar. 8, no. 3--4 (September 1977), 197--211, received 29 January 1976. The conjecture is Erdős's, stated in 1965 and again in 1973 as never proved. The site's curator, Thomas F. Bloom, marks Problem 362 proved and credits this paper with the affirmative answer to the second question, as a consequence of its multidimensional theorem. The page is dated by the first day of the issue month, since the paper's first posting carries no finer date.

Depends on. Halász 1977, Theorem 2, whose result page details the reduction to the fixed-size counts.