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These inputs are external to this paper and their full proofs are not reconstructed in this source unit.

• Lemma 3 is the Berry–Esseen inequality with independent centered summands, positive total variance, and finite third absolute moments.
• Theorem 3 records the exact positive-input CFP theorem, including the distinction between its retained set and its subset-sum witness.
• The standard divisor estimate used on published p. 8 is: there is an absolute $C>0$ such that, for all sufficiently large $q$ and integers $1\le a\le q^2$,

$$\tau(a)\le q^{C/\log\log q}.$$

• The near-one smooth-number estimate used on p. 9 is: for each fixed $u\in(1/2,1)$,

$$|\{m\le n:m\text{ is }n^u\text{-smooth}\}| =(1+\log u+o_u(1))n.$$

This is the corresponding Dickman asymptotic, with the reciprocal parameter convention made explicit.

• The prime number theorem, in the form $\pi(t)\sim t/\log t$, supplies $\pi(t)=o(t)$ as $t\to\infty$.
• The $r=1$ case of Croot's short-interval theorem says that for every sufficiently large integer $N$ there are distinct $N<b_1<\cdots<b_\ell\le(e+o(1))N$ with $\sum_i1/b_i=1$. The retained arXiv:math/9904181v1 statement on p. 1 gives $(e^r+O_r(\log\log N/\log N))N$ for each fixed rational $r>0$. This unit uses only $r=1$. It does not identify that manuscript bytewise with the published Acta Arithmetica 99 (2001), 99–114 article.

Finite entropy identities used in the source are expanded locally in entropy basics. Calculus, finite probability, integer factorization, and elementary finite counting are used directly.

Source: published PDF, pp. 4, 7–10 and references [1], [5]–[7], [10]. No original Dickman, prime number theorem, divisor-bound, or Berry–Esseen proof is included. A later paper's use of these results is not a substitute for such a proof.

Bears on. Problem 297.