Larcher's finite-prefix interface and qualitative ratio growth

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The mean-normalized target of Problem 1221 permits repeated points. The [[research/erdos_1221/connections|existing multiset extension]] and closing reconstruction rest on the span-to-count argument and, as the preprint argues, on two imported discrepancy theorems. This page records what Larcher's paper supplies for the first import and two ways to do without its optimized constant. The composed argument now carries the whole chain for all sequences, with the elementary finite-list bound in place of the first import. This source-interface analysis and the constant calculation below are author-recorded by Plasma AI; they supply no independent acceptance of the full chain.

What the source actually states

Section 3 of Gerhard Larcher's [[../library/analysis/larcher_2014_star_discrepancy_sequences_unit_interval/_index|On the star discrepancy of sequences in the unit interval]], arXiv:1407.2094v1, starts with an arbitrary finite list $x_1,\ldots,x_N\in[0,1)$, $N=\lfloor a^t\rfloor$, $3<a<4$. Writing $D_j(u)=\#\{i\le j:x_i<u\}-ju$, it applies the auxiliary lemmas to the integral of $\max_{j\le N}D_j-\min_{j\le N}D_j$ and concludes, before passing to an infinite sequence, that some $j\le N$ has

$$\sup_u|D_j(u)|\ge c_a\log N, \qquad c_a=\frac{(a-2)(8a+3)}{16(1-2a)^2\log a}.$$

Thus the required finite-list quantifier is present in the proof itself; it is not inferred merely from the abstract's infinitely-many-prefixes theorem. Section 3 and the definitions at the start of Section 2 have been inspected in the source TeX. The complete optimization proof of the auxiliary lemmas has not been checked. The text writes block endpoints such as $a^{t-1}$ and the equality $t=\log N/\log a$ while suppressing integer-rounding adjustments, and counts distinct jump locations. These points need explicit handling in a fully checked derivation. No discrepancy constant is newly certified here.

Under the finite-list bound as stated, any list of length $L$ has $H_L\ge c_a\log L-O_a(1)$: keep the prefix of the largest $\lfloor a^t\rfloor\le L$. Its logarithm differs from $\log L$ by $O_a(1)$, and the maximum over prefixes cannot increase when prefixes are removed. At $a=7/2$, the source's expression is $31/(384\log(7/2))>1/16$, so an absolute large-$L$ threshold gives the constant used in the existing reconstruction, provided the source's finite-list estimate is established with an additive error bounded uniformly in the list. Repeated coordinates are covered by the ordered-perturbation argument in Proposition 1221.1 of the connections page: for every $\eta>0$ a distinct list has prefix error at most $H_L+\eta$, and then $\eta\downarrow0$ transfers a universal lower bound.

The qualitative ratio needs only a positive constant

Suppose, more weakly, that every finite list satisfies $H_L\ge c_0\log L-C_0$, with absolute $c_0>0$ and $C_0$. Proposition FP of the finite-list page proves this with $c_0=1/(16\log4)$ and $C_0=1/16$, by Larcher's range splitting and a quadratic potential, without his optimized lemmas. The span-to-count estimate and its reading for multisets are checked on the composition page. Set $A=\alpha\log r$ with $0<\alpha<c_0/6$. The ratio contradiction then gives

$$B=3A+O\left(\frac A{\log(r/A)}\right) =3\alpha\log r+O(1), \qquad S=\frac{\sqrt{Ar}}{\log^2(r/A)}.$$

The multiset extraction gives a list of length $L'\in[\lfloor S\rfloor-B,\lfloor S\rfloor+B]$ with $H_{L'}\le B$. Since $B/S\to0$,

$$\log(\lfloor S\rfloor-B) =\tfrac12\log r-\tfrac32\log\log r+O(1).$$

Consequently $3\alpha\log r+O(1)\ge(c_0/2)\log r-O(\log\log r)$, a contradiction. All constants are independent of the original sequence. This yields $\mu_r-1\ge\alpha\log r/r$ for large $r$, hence $r(\mu_r-1)\to\infty$; with Proposition FP, $\alpha=1/200$ is admissible, and the composition page writes the comparison out. The two one-sided divergences need the separate planar $L^1$ input, which the composition page takes from the written-out Halász bound.

A qualitative finite-prefix bound from a held source

For the limit $r(\mu_r-1)\to\infty$, even a logarithmic discrepancy rate is unnecessary. The following compactness argument uses the [[../library/discrepancy/schmidt_1972_irregularities_distribution/corollary_p64|countability corollary of Schmidt's 1972 part VI]], rather than the different planar theorem of part VII. For every infinite sequence in $(0,1]$, that corollary says the set of $u$ at which the anchored counting error stays bounded is countable. The definitions and corollary were checked against Section 1 of the held transcription, printed pp. 63--64. Its published proof is an external premise, not independently reverified here.

Uniform qualitative bound. For every finite $K\ge0$ there is an integer $L_K$ such that every list $z_1,\ldots,z_L\in[0,1]$ with $L\ge L_K$ has $H_L>K$, allowing repeated entries. Here $H_L$ has the strict counting convention of the finite-prefix interface.

Proof. For a prefix of length $j$, let $z_{(1)}\le\cdots\le z_{(j)}$ be its order statistics, retaining multiplicity. Its star discrepancy in counting units is

$$\sup_{0\le u\le1} \left|\#\{i\le j:z_i<u\}-ju\right| =\max_{1\le i\le j} \max\{i-jz_{(i)},\ jz_{(i)}-(i-1)\}.$$

The left and right limits at each distinct coordinate give the two extrema; within a repeated block the last and first indices, respectively, give the extremal values. At $0$ and $1$ the same formula holds, with the unused one-sided value contributing only zero. Order statistics are continuous in the coordinates. Thus $H_j$, the maximum of this expression over its first $j$ prefixes, is continuous on $[0,1]^j$.

Suppose arbitrarily long finite lists had $H_L\le K$. In the compact product $[0,1]^{\mathbb N}$, put

$$\mathcal C_L=\{z:H_L(z_1,\ldots,z_L)\le K\}.$$

These sets are closed, nested and nonempty: any sufficiently long list supplies the needed prefix and can be extended arbitrarily. Compactness gives an infinite sequence in their intersection, with every prefix discrepancy at most $K$. In any prefix, letting $u\downarrow0$ shows that its number of zero entries is at most $K$. Hence the whole sequence contains at most $\lfloor K\rfloor$ zero entries. Replace them by $1$. This produces a sequence in $(0,1]$ and changes every prefix counting error by at most $\lfloor K\rfloor$. For this sequence every $u\in(0,1]$ therefore has error bounded by $2K$ at every prefix, contradicting Schmidt's countability corollary. This proves the uniform bound. $\square$

This is an author-recorded deduction by Plasma AI, with no effective estimate for $L_K$. It does not certify Larcher's constant or rounding argument. The composed argument uses Proposition FP instead, so this bound is an independent alternative resting on Schmidt's published theorem.

Closing the qualitative ratio with a fixed parameter

This uses the all-sequence span-to-count implication of Proposition 3.1 and the multiset extraction in Proposition 1221.1 of the connections page, read for coincident terms on the composition page. The latter supplies a list of length $L'$ with $H_{L'}\le B$ and $|L'-\lfloor S\rfloor|\le B$. These inputs keep the standing the composition page records.

Fix any real $A\ge1$. If for some sequence $\limsup_n M_n^{(r)}/m_n^{(r)}<1+A/r$, choose $0<C<A$ so that the ratio is at most $1+C/r$ for all sufficiently large $n$. The mean-span identity $nm_n^{(r)}\le r\le nM_n^{(r)}$ then gives

$$n(M_n^{(r)}-m_n^{(r)})\le C,\qquad nM_n^{(r)}\le r+C.$$

With $n=\lfloor t\rfloor$, the nonnegative choices $a_t=r-nm_n^{(r)}$ and $b_t=tM_n^{(r)}-r$ satisfy $a_t+b_t\le C+(r+C)/n<A$ eventually. Thus the same real-time span hypothesis used in the closing reconstruction holds, including for coincident terms as read on the composition page. Proposition 3.1 gives

$$B=3A+\frac{C_1A}{\log(r/A)},\qquad S=\frac{\sqrt{Ar}}{\log^2(r/A)}.$$

As $r\to\infty$ with $A$ fixed, $B\le4A$ eventually and $S\to\infty$. The extracted list consequently has $L'\ge\lfloor S\rfloor-B\to\infty$ and $H_{L'}\le4A$. This contradicts the uniform qualitative bound with $K=4A$. Its length threshold depends on $A$ but not on the sequence, so for every fixed $A$ and all sufficiently large $r$, no sequence has the strict ratio bound above. Taking the infimum over sequences yields $r(\mu_r-1)\ge A$. Since $A$ was arbitrary, the ratio expression tends to infinity, with the standing of these inputs.

The eventual time at which a sequence obeys the count bound may depend on that sequence; the contradiction only needs its existence. The threshold in $r$ does not depend on the sequence. This separates the two quantifiers and avoids a uniformity claim about convergence times.

This qualitative route is independent of both Larcher's optimized minimization and Proposition FP, but rests on Schmidt's published theorem. It gives no logarithmic rate, no coefficient $1/100$, and neither one-sided divergence. The rate $\mu_r-1\ge\log r/(200r)$ follows instead from Proposition FP, and the one-sided conclusions from the planar $L^1$ bound, whose statement in counting normalization is Bilyk's [[../library/discrepancy/bilyk_2011_roth_orthogonal_function_method/equation_1_14|exposition of Halász's bound]] and whose proof is written out on the planar page; both are composed on the composition page. Larcher's optimized lemmas would raise the ratio coefficient to Korsky's $1/100$, and they remain unchecked.