Prime-multiple blocks longer than three times the largest prime

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Target and conventions. For Problem 1143, let $G_\alpha(u)$ minimize once-counted prime-multiple locations over all sets of $u$ distinct primes with largest member $P$ and all blocks of $\lfloor\alpha P\rfloor$ consecutive positive integers. This account studies $3\le\alpha<4$. It does not determine the minimum for a prescribed prime set, and the catalog problem remains open.

The three-position test has two different outcomes. Separated repeated prime grids incur a sharp linear cost. Even without separated rows, full Cartesian grids retaining their row and column locations incur at least a constant times $u^{2/3}$. Nonrectangular cell sets avoid these obstructions: an explicit rational measure on Lemm's seven-point configuration gives $G_\alpha(u)\le C u^{0.56213}$ throughout $3\le\alpha<4$, with $C$ absolute. All project arguments below are recorded at tier 0, without independent review.

Source boundaries. The author-recorded [[theory/number_theory/L37_prime_multiple_block_extremum/_index|L37 gives G_alpha(u)=ceil(2 sqrt(u)) for 2<alpha<3]], extending the Erdős--Selfridge square construction; the short-block account keeps its scope explicit. For longer intervals, [[../library/integer_sequences/erdos_1986_problems_number_theory/theorem_p62|Erdős's p. 62 theorem]] gives $(6u)^{1/2}$ for square $u$ when the real interval length is at least $3P$. A block of $K\ge3P+1$ integers meets that printed length convention; this quotation is not used to settle the endpoint $K=3P$. [[../library/primes/ruzsa_1995_few_multiples_many_primes/theorem|Ruzsa's 1995 theorem]] gives an upper bound $O_\alpha((u\log u)^{2/3})$ in the present range. His constructed block has at least $\alpha P$ integers, so taking a subblock gives the convention used here. [[../library/primes/green_2017_arithmetic_kakeya_conjecture_katz_tao/proposition_4_1|Green and Ruzsa's Proposition 4.1]] compares the integer-ratio minimum with arithmetic Kakeya sets. None of these results gives a formula for each prescribed prime set.

The exact three-layer model

Index primes by integer cells $(x,y)$ with distinct differences $y-x$, the prime of a cell being $q+y-x$. Choose $z\equiv-x$ modulo each prime. Its multiples are

$$z+tq+(1-t)x+ty\qquad(t\in\mathbb Z).$$

The three central layers are $z+x$, $z+q+y$ and $z+2q+2y-x$. L198 proves that, if $|x|,|y|\le\omega$ and $(4-\alpha)q\ge16\omega+1$, a positive block of exactly $\lfloor\alpha P\rfloor$ integers contains precisely those layers. Their disjointness makes its count exactly $|X|+|Y|+|W|$, where $X,Y,W$ are the value sets of $x,y,2y-x$. Multiple divisibility within a layer is counted once.

L37's grid $q+a_i+b_j$ corresponds to $x=-a_i$, $y=b_j$. The old two positions are the row $z-a_i$ and column $z+q+b_j$. A third consecutive multiple is forced once the block has $3P$ integers. The right extension adds $z+2q+a_i+2b_j$; the left extension adds $z-q-2a_i-b_j$. Counting only the old row and column positions therefore misses an entire forced set.

Separated grids have a sharp linear cost

In the repeated groups used in L37, $a_{i+1}-a_i>b_s$. Under the narrow condition $2a_r+3b_s<q$, and with $K=\lfloor\alpha P\rfloor<4q$, [[theory/number_theory/L199_separated_grid_third_layer/_index|L199 gives the exact minimum over blocks retaining the row and column positions]]:

$$r_S+s_S+|T_S|,\qquad T_S=\{a_i+2b_j:(i,j)\in S\}.$$

Here $S$ is any selected set of grid cells and $r_S,s_S$ count its occupied rows and columns. Every value in $T_S$ has at most two cells, so $|T_S|\ge\lceil|S|/2\rceil$. The canonical positive translate of $[z-q+1,z-q+K]$ attains the minimum. Moving it while retaining the core cannot help: every missing following location forces a distinct preceding location into the block. L199 also proves a linear lower bound for all blocks in the specified window around $z$.

For a full $r\times s$ separated grid,

$$|T_S|\ge rs-(r-1)\lfloor s/2\rfloor.$$

This is sharp for every $r,s$ and every $3\le\alpha<4$. Columns $0,1,\ldots,s-1$ and rows spaced $2\lceil s/2\rceil$ apart attain the projection bound; the prime-realization theorem below realizes a dilate of this pattern with all the endpoint hypotheses. Its core-retaining minimum is exactly $r+s+rs-(r-1)\lfloor s/2\rfloor$.

This conclusion fixes the CRT placement and the stated block families. Distant blocks for the same primes can realize other residue patterns; no lower bound for all such blocks follows.

Full Cartesian grids also have an obstruction

Dropping row separation does not make full rectangles competitive. For finite sets $A,B\subset\mathbb Z$ with unique sums $a+b$, put $r=|A|$, $s=|B|$. Then L204 proves

$$|A+2\cdot B|\ge s\sqrt r,\qquad |2\cdot A+B|\ge r\sqrt s.$$

The dot denotes dilation. To see why sumset growth applies, take a subset $X\subseteq2\cdot B$ with minimal growth under adding $A$. Petridis's lemma bounds $|X+A+A|$ by $(|A+2\cdot B|/s)^2|X|$, while the unique sums force its subset $X+2\cdot A$ to have $r|X|$ elements. The complete proof is in L204.

For a narrow full grid of $u=rs$ distinct primes, a median cut chooses a large subrectangle whose following multiples are all present, or whose preceding multiples are all present. Thus every block with $K\ge3P$ retaining all row and column locations has count at least

$$r+s+\frac{\min(s\sqrt r,r\sqrt s)}{2\sqrt2} \ \ge\ \frac{3}{2^{5/3}}u^{2/3}.$$

This includes mixtures of the preceding and following layers. It rules out full Cartesian grids as an asymptotically optimal family in this range. It does not assert that $2/3$ is the optimal exponent even within that family.

Deleting $v$ cells removes at most $v$ values from a projection. Thus for a subset $S\subseteq A\times B$ with $u=rs-v$ cells, its canonical following layer still has at least $s\sqrt r-v$ locations. In particular, nearly full balanced rectangles with $r,s\asymp\sqrt u$ and $v=O(\sqrt u)$ pay order $u^{3/4}$ in that layer. The argument allows sparse, nonrectangular selections to behave differently.

A useful comparison family is the base-six product of rows $\{0,2,4\}$ and columns $\{0,1\}$ in cell coordinates. It has $6^n$ cells with distinct differences and at most $4^n$ third-projection values, giving exponent $\log4/\log6\approx0.774$ after prime realization. The universal rectangular lower exponent and this explicit upper exponent do not match; neither approaches the nonrectangular bound below.

Prime realization and arithmetic Kakeya

The exact prime-set comparison holds across the whole half-open range. Define $F'_3(u)$ using finite integer sets containing three-term progressions with $u$ distinct positive common differences. Then L202 proves

$$F'_3(u)\le G_\alpha(u)\le3F'_3(u) \qquad(3\le\alpha<4,\ u\ge1).$$

For the lower bound choose three consecutive multiples of every prime. For the upper bound turn a progression $a,a+d,a+2d$ into the cell $(a,a+d)$. All three projections lie in the progression-containing set. Every finite cell set can be realized without discarding any cell: use a sufficiently late segment of a much longer prime progression, with its common difference as the cell dilation. Green--Tao's ordinary arbitrarily-long-progression theorem supplies it. The construction makes $q$ large enough relative to every dilated coordinate for L198; no additional short-interval progression theorem is needed.

The source statement used is Theorem 1.1, p. 2, of Green and Tao, arXiv:math/0404188v6, checked on the page image on 2026-10-15. Its proof is an external premise. The deduction extends Green--Ruzsa Proposition 4.1's construction to noninteger ratios, not a new Green--Tao theorem. It also shows $G_3(u)\le G_\alpha(u)\le3G_3(u)$, with constants uniform below four.

Nonrectangular type classes and the remaining exponent

For a cell set $S$ let $N(S)=\max(|X(S)|,|Y(S)|,|W(S)|)$ and let

$$\beta=\sup_S\frac{\log|S|}{\log N(S)},$$

where $|S|\ge2$ and $y-x$ is injective. The author-recorded L200 proves

$$G_\alpha(u)\ge u^{1/\beta}\quad(\alpha\ge3),\qquad \lim_{u\to\infty}\frac{\log G_\alpha(u)}{\log u}=\frac1\beta \quad(3\le\alpha<4).$$

Its proof uses tensor powers and prime-number-theorem averaging, which loses only a logarithmic factor. The change of variables $(a,b)=((2y-x)/2,x/2)$ identifies $\beta$ with the sums--differences constant for input slopes $0,1,\infty$ and output slope $-1$. Determining this constant is the missing independent progression estimate, not a consequence of the prime construction.

The cells $(0,1),(2,1),(2,2)$ yield a first explicit improvement over rectangles. Words using each cell exactly $m$ times have $(3m)!/(m!)^3$ distinct differences and exactly $\binom{3m}{m}$ values in each projection. Hence L201 gives

$$G_\alpha(u)\le C u^\theta(\log u)^{\theta-1/2},\qquad \theta=\frac{\log(27/4)}{\log27}=0.57938\ldots.$$

The exponent comes from Ruzsa's three-point example and the entropy-to-type-class method recorded by Lemm. L201 also retains a proof using only the prime number theorem, with bound $O_\alpha(u^\theta(\log u)^2)$. The complete type classes themselves have a residual logarithmic factor; realizing all their differences as primes does not prove $O(u^\theta)$.

Using the seven-point configuration from Lemm's Theorem 2.1 instead, L203 gives

$$G_\alpha(u)\le C u^{56213/100000}, \qquad 3\le\alpha<4,$$

with $C$ absolute. Its rational measure has joint entropy $H$ and largest projection entropy $h$ with $h/H<0.56213$. Exact multinomial counts, not finite searches, give the unbounded family. The numerical inequality is certified by rational logarithm enclosures. Lemm retains authorship of the configuration; this is a rigorous prime-block application, with no claim of a new sums--differences record.

For comparison, Katz and Tao, Bounds on arithmetic projections, and applications to the Kakeya conjecture, Math. Res. Lett. 6 (1999), 625--630, arXiv:math/9906097v3, Theorem 1.1, bounds the number of differences by $N^{11/6}$ when the three input projections have size at most $N$. The statement and its hypotheses were checked on the p. 1 image on 2026-10-15; the source proof is not reconstructed here. Applied to the cell change of variables it gives $\beta\le11/6$, and therefore the known lower bound $G_\alpha(u)\ge u^{6/11}$ for every $\alpha\ge3$. Together with L203,

$$u^{6/11}\le G_\alpha(u)\le C u^{0.56213} \qquad(3\le\alpha<4).$$

The library's Theorem 2.1 page separates Lemm's claimed optimized ratio from the weaker ratio of the printed rounded probabilities. Its Proposition 1.1 page also records the limiting qualification in the entropy-to-set passage. L203 uses its own exact rational witness and a strict entropy margin, so neither qualification is silently removed. The three-slope interval is also recorded in Section 1.1 of Tao, arXiv:2511.15135v1. Only that introductory comparison was used; neither this account nor its narrow literature search establishes a new record.

Sketch, standing and precise gap

The settled author-recorded chain is L198 for exact locations, L202 for full prime realization, and L201 or L203 for the cell construction. L199 settles the separated-grid optimum, using L202 only for its prime sharpness examples. L204 rules out every full Cartesian grid retaining its core at the scale achieved by L203. L200 identifies the unrestricted asymptotic obstruction as the sums--differences constant. All these project deductions are credited to Plasma AI at tier 0. L37 is contextual and is not consumed by their proofs.

The remaining gap is an estimate for arbitrary nonrectangular cell sets: either a configuration with a better projection-to-joint entropy ratio, or a stronger universal three-projection inequality. The exact value of $\beta$, the exact finite function $F'_3(u)$, and any sharper comparison than the factor three above are not settled. Improving the prime realization alone cannot determine the exponent, because every finite cell pattern already has such a realization. An argument for prescribed primes would need additional information about that prime set. Ratios $\alpha\ge4$ require further layers and are outside this three-position argument.

The [[theory/number_theory/L198_three_layer_prime_blocks/evidence/_index|L198 finite checks]] test actual positive blocks and once-only counting; one contains 220 multiples of 302 primes at $\alpha=3$. [[theory/number_theory/L199_separated_grid_third_layer/evidence/_index|L199's checks]] enumerate its two block families for small prime grids. [[theory/number_theory/L203_rational_entropy_prime_block_bound/evidence/_index|L203's certificate]] checks the exact entropy witness. The universal assertions rest on their written proofs and stated source premises. None of these author checks supplies independent acceptance or changes the general problem's standing.