Proper-subsum sparsity does not force holes

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The final section records a different, arithmetic obstruction to a local-repair covering argument: almost every pair of nearby targets lacks representations with sufficiently close physical root vectors. It too does not force holes.

Status and scope

The interval bounds below are proved. The Cassels basis theorem is the external input; its statement and proof were checked in Nathanson, Cassels Bases, Theorem 10. The resulting counterexample strengthens the elementary convex-track warning. It rules out a general covering argument based only on counting regularity, finite representation mean, and proper-subsum anti-concentration. It is not a counterexample about powerful numbers.

An interval bound from bounded-error counting

Suppose $r\ge3$ and $A\subset\mathbb N$ satisfies

$$A(t):=\#(A\cap[1,t])=c\,t^{1/r}+O(1),\qquad c>0.$$

For $1\le k\le r$ and $1\le H\le X$, let $T_{A,k}(X,H)$ count ordered tuples in $A^k$ with sum in $[X,X+H]$. Then

$$T_{A,k}(X,H)\ll_{A,r,k} X^{(k-1)/r}+H X^{k/r-1}. \tag{1}$$

Every counted tuple has an entry at least $X/k$. Fix its index and the other $k-1$ entries, allowing at most $O(X^{(k-1)/r})$ choices. The remaining entry lies in an interval of length at most $H$ contained in $[X/k,2X]$. The counting hypothesis and the mean value theorem bound its possibilities by $O(1+H X^{1/r-1})$. Summing over the possible indices proves (1).

An actual basis with this regularity

Cassels' theorem supplies, for every $r\ge3$, an additive basis $C=\{c_n\}\subset\mathbb N_0$ of order $r$ with

$$c_n=\gamma n^r+O(n^{r-1}),\qquad \gamma>0.$$

Inverting this estimate gives $C(t)=\gamma^{-1/r}t^{1/r}+O(1)$. After replacing $C$ by $A=C+1$, all its terms are positive, $rA$ is cofinite, and the same counting estimate holds. Thus (1) applies to an actual additive basis.

In particular, at the natural root-gap scale $H=X^{1-1/r}$,

$$T_{A,r-1}(X,H)\ll X^{1-2/r}=H X^{-1/r}. \tag{2}$$

Every such interval contains only a vanishing proportion of $(r-1)$-fold sums, although adding one more copy of $A$ covers all sufficiently large integers.

These bases also have finite limiting unordered representation mean:

$$\frac1X\sum_{n\le X}R_{A,r}(n) \longrightarrow \frac{c^r\Gamma(1+1/r)^r}{r!}, \qquad c=\gamma^{-1/r}.$$

Indeed, scaled counting measures from $A(t)\sim ct^{1/r}$ give the usual simplex integral for ordered tuples. Its constant is $c^r\Gamma(1+1/r)^r$, since $r(1/r)=1$. Tuples with repeated entries number $O(X^{1-1/r})$, so division by $r!$ gives the unordered constant.

Comparison with powerful sums

For a finite canonical-core set $D$,

$$A_D=\bigcup_{d\in D}\{d m^r:m\ge1\}$$

has disjoint tracks, by canonical uniqueness. Therefore

$$A_D(t)=\left(\sum_{d\in D}d^{-1/r}\right)t^{1/r}+O_D(1).$$

Thus the bounded-error counting property of these finite unions, and all the bounds (1), are shared by the Cassels example.

For the full powerful set $\mathcal P_r$, the established core count and tail give

$$\begin{aligned} \mathcal P_r(t) &=\sum_{\substack{d\in\mathcal C_r\\d\le t}} \left\lfloor(t/d)^{1/r}\right\rfloor\\ &=W_rt^{1/r}+O_r(t^{1/(r+1)}). \end{aligned}$$

The floor errors are bounded by the core count, and the missing weight contributes $t^{1/r}T_r(t)=O(t^{1/(r+1)})$. The same interval argument now yields

$$T_{\mathcal P_r,k}(X,H) \ll_{r,k}H X^{k/r-1}+X^{(k-1)/r+1/(r+1)}. \tag{3}$$

For $k=r-1$ and $H=X^{1-1/r}$, in particular,

$$\#\bigl((r-1)\mathcal P_r\cap[X,X+H]\bigr) \ll_r H X^{-1/[r(r+1)]}. \tag{4}$$

This is genuine local sparsity of proper subsums. But (2) is stronger and holds for an additive basis, so it cannot by itself force missing targets.

What the attempt rules out

No universal gap criterion can use only the above counting regularity, finite mean, and gap-scale proper-subsum upper bounds. This counterexample does not have multiplicative closure and is not an exact union of the root lattices $\{d^{1/r}m:m\in\mathbb N\}$. It also does not prescribe the particular numerical mean of the powerful representation function. An argument exploiting those additional features is not excluded. No such argument was established in this continuation.

An arithmetic obstruction to local repair

For a representation $n=\sum_i a_i$, call $(a_1^{1/r},\ldots,a_r^{1/r})$ its physical root vector, padding with zeros when there are fewer summands and allowing permutations when comparing two representations.

Proposition. Fix $r\ge3$, a nonzero integer $h$, and $\eta>0$. Put $B=X^{1/r}$. Only $o(X)$ integers $n\le X$ admit powerful-sum representations of both $n$ and $n+h$ whose matched physical root vectors have sup-distance at most $B^{1/(r-1)-\eta}$.

This is an all-core statement about targets, not a bound for the number of all representation pairs. It rules out repairing almost every target by changing its roots by this amount. A cofinite cover whose representations make larger jumps is not excluded.

A uniform binary bound for fixed coefficients

For fixed positive integers $d,e$ and $r\ge3$, uniformly for $c\ne0$, $|c|\ll B^r$,

$$\#\{1\le x,y\ll B:dx^r-ey^r=c\} \ll_{d,e,r,\epsilon}B^\epsilon. \tag{5}$$

Here and below the constants implicit in the ranges are fixed.

If $(e/d)^{1/r}=a/b\in\mathbb Q$ in lowest terms, then $d=kb^r$, $e=ka^r$ for an integer $k$. When solutions exist, $(bx)^r-(ay)^r=c/k$ is an integer equation. The nonzero difference $bx-ay$ divides $c/k$. Fixing that divisor leaves a nonconstant polynomial of degree $r-1$ in $ay$, so the divisor bound proves (5).

Otherwise let $\gamma=d(e/d)^{1/r}$ and $K=\mathbb Q(\gamma)$, of degree $m\ge2$. The element $\gamma$ is integral, and its monic minimal polynomial $P(T)$ divides $T^r-ed^{r-1}$ in $\mathbb Z[T]$. Homogenizing that factorization proves

$$0\ne N_{K/\mathbb Q}(dx-\gamma y) =y^mP(dx/y)\mid d^{r-1}c. \tag{6}$$

This remains true if the degree-$r$ binomial is reducible.

There are at most $\tau(|d^{r-1}c|)^m$ integral ideals whose norms divide $|d^{r-1}c|$: at a rational prime of exponent $v$, there are at most $m$ prime ideals above it, each with exponent at most $v$. For any solution $\beta=dx-\gamma y$, its conjugates satisfy

$$(CB)^{-(m-1)}\le|\sigma(\beta)|\le CB.$$

The lower bound uses its nonzero integral norm. For each occurring principal ideal choose one solution-generator $\beta_0$. Every other one is $\varepsilon\beta_0$, where the unit logarithms lie in a box of side $O(\log B)$. The logarithmic unit image is discrete, with finite kernel: bounded logarithms bound all conjugates, and only finitely many algebraic integers of a fixed degree have all conjugates bounded. The image lies in a hyperplane of dimension at most $m-1$. Packing gives $O_K((1+\log B)^{m-1})$ possible generators, uniformly in the ideal. Finally $(x,y)\mapsto dx-\gamma y$ is injective. The divisor bound, with its exponent chosen smaller initially, absorbs these logarithms and proves (5).

Counting nearby pairs and removing the core cutoff

Fix two canonical coefficient tuples $\mathbf d,\mathbf e$. Suppose the outputs differ by $h\ne0$ and

$$|d_i^{1/r}x_i-e_i^{1/r}y_i|\le H\quad(1\le i\le r).$$

At least one coordinate has $d_jx_j^r-e_jy_j^r\ne0$. Choose such a coordinate and fix the other $r-1$ coordinate pairs; each pair has $O_{\mathbf d,\mathbf e}(B(H+1))$ possibilities. The last coordinate satisfies a nonzero equation covered by (5). Thus the number of pairs for these fixed cores is

$$O_{\mathbf d,\mathbf e,r,\epsilon} \bigl(B^{r-1+\epsilon}(H+1)^{r-1}\bigr). \tag{7}$$

For $H=B^{1/(r-1)-\eta}$ this is $o(B^r)$ after choosing $\epsilon>0$ sufficiently small. The nonzero-coordinate condition is essential: (5) is false for $c=0$.

Now fix a core cutoff $Y$. The finitely many core tuples and coordinate permutations contribute $o_Y(X)$ targets. Targets having any representation with a core exceeding $Y$ number at most $O_r(XT_r(Y))$, by the established canonical first-moment tail; the same bound applies after the fixed shift $h$. Shorter representations contribute $O_r(X^{1-1/r})$ targets. Hence the upper density in the proposition is $O_r(T_r(Y))$. Let $Y\to\infty$ to prove the claim. No uniform power-saving rate after summing all cores is asserted.

Why the covering argument does not close

There is no theorem making a cofinite cover continuous in root space. Simple cap pigeonholing does not supply one at the required scale. For targets comparable to $B^r$, a block of $L$ targets lies in an $\ell^r$-shell of thickness $O(L/B^{r-1})$. If this is at most $H$, the shell has $O_r((B/H)^{r-1})$ caps of diameter $O(H)$. Forcing a positive fraction of close pairs by overfilling these caps requires

$$L H^{r-1}\gg B^{r-1}.$$

By contrast, summing (7) over differences $1\le|h|\le L$ makes the nearby-pair count $o(B^r)$ only in the sufficient range $L(H+1)^{r-1}\ll B^{1-\epsilon}$. The scales are separated by $B^{r-2}$. This defeats that particular pigeonhole argument, not every use of exact root geometry. No required root-coherence theorem or higher-exponent gap follows.