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Let $K$ be a fixed positive integer. Choose a positive integer $N_0$ so that the $r=1$ Croot short-interval input gives a set of distinct denominators in $(N,3N]$ summing to 1 for every integer $N\ge N_0$. If $z_f>0$ has denominator dividing $K$ and

$$z_f\le x\le\xi\log n,\qquad K\xi\log4\le1/2,$$

then, for all sufficiently large $n$ depending only on $K,N_0$, there is a set $A_2\subseteq\{a\in[n]:K\mid a\}$ with $s(A_2)=z_f$.

Source: published PDF, p. 10. This gives integer interval endpoints and a uniform size bound for the printed terminal step. Croot's theorem itself is external.

Bears on. Problem 297.

Proof

The number $m=Kz_f$ is a positive integer and $m\le Kx\le K\xi\log n$. For $j=0,\ldots,m-1$, put $N_j=N_0\,4^j$ and choose a Croot representation of 1 with denominators in $(N_j,3N_j]$. These intervals are disjoint, since $3N_j<N_{j+1}$. The union $D$ of the representations has distinct denominators and $s(D)=m$. Every denominator in it is at most

$$3N_0\,4^m\le3N_0\,n^{K\xi\log4}\le3N_0\sqrt n.$$

For all sufficiently large $n$ this is at most $n/K$. Set $A_2=KD$. Its elements are distinct multiples of $K$ in $[n]$, and $s(A_2)=s(D)/K=m/K=z_f$, as required.