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Use the parameters and positive coefficients from [[additive_combinatorics/adamczewski_2026_erdos1/normal_coefficients|the normal-coefficient construction]], so $q_0=2^{s+r}$ and

$$0<a_i(t)\leq2D(2^s)^n\qquad(0\leq i\leq n). \tag{1}$$

For every labeled pair

$$(i,j),\qquad 0\leq i\leq n,\quad 0\leq j<s+r,$$

form the positive integer

$$b_{i,j}=2^ja_i(t). \tag{2}$$

Distinct elements and subset sums

First the labeled weights in (2) are pairwise distinct. If $b_{i,j}=b_{i',j'}$, use the two digit vectors having respectively the single nonzero digits $2^j$ in coordinate $i$ and $2^{j'}$ in coordinate $i'$. Both digits are smaller than $q_0$. Their images under the map of [[additive_combinatorics/adamczewski_2026_erdos1/digit_injectivity|digit injectivity]] agree, so the digit vectors agree. Hence $i=i'$ and $j=j'$.

We may therefore define the set

$$A=\{b_{i,j}:0\leq i\leq n,\ 0\leq j<s+r\},$$

and its cardinality is exactly

$$|A|=(n+1)(s+r). \tag{3}$$

A subset of $A$ chooses bits $\varepsilon_{i,j}\in\{0,1\}$. For each $i$ let

$$x_i=\sum_{j=0}^{s+r-1}\varepsilon_{i,j}2^j,$$

so $0\leq x_i<q_0$, and the subset sum is $\sum_i x_i a_i(t)$. Different subsets give different bit arrays because the labeled weights are distinct; uniqueness of binary expansion makes their digit vectors different. Digit injectivity then makes their subset sums different. Thus $A$ is sum-distinct.

Range and ratio

From (1), every element of $A$ is at most

$$2^{s+r-1}\,2D(2^s)^n =D2^{(n+1)s+r}.$$

Set

$$N=D2^{(n+1)s+r}. \tag{4}$$

Then $A\subseteq\{1,\ldots,N\}$, and (3)–(4) give the exact cancellation

$$\frac{2^{|A|}}{N} =\frac{2^{(n+1)(s+r)}}{D2^{(n+1)s+r}} =\frac{2^{nr}}{D}. \tag{5}$$

The lattice reduction supplied $kD<2^{nr}$, so (5) yields

$$kN<2^{|A|}.$$

Source and dependencies

An explanation of the proof of Erdős Problem 1, preliminary exposition with no named author (erdosproblems.com, 2026), §7, equations (26)–(28), pp. 8–9. The edition read is named on the source card. The proof establishes pairwise distinctness of the labeled weights before treating them as a set; this is needed for the cardinality and subset encoding in (3).

Bears on. #1.