The critical exponent for isoperimetric inequality on the hamming cube

Upper bound, last-listed row

| $0.5$ | [DIRX2026] | Solves the problem by establishing $\beta=0.5$. |tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $0.5$ | [BIM2023] | For every $\beta<1/2$, Hamming ball examples give half-size sets with $\mathbb{E}h_{A}(x)^\beta$ arbitrarily small as $n\to\infty$ |tracked since 2026-07-24 — no movement seen yet

Checked against material for its cited source

• The row shown above: bound row $0.5$ line 36 citing [BIM2023] — the claim we checked: that the cited paper gives the Hamming-ball construction showing the exponent 0.5 in the row's sense: for every β below 1/2, sets of half the cube's size with the stated expectation arbitrarily small. What we read could not settle it. source link (read in the body: arXiv v1 (the only version, and the edition the entry cites), eq. (1.4) on p. 2 and the Section 3 appendix on p. 16, read from page images. The appendix computes E h_A^β for an even-dimensional ball slightly LARGER than half the cube; it separately uses an exactly half-size odd-dimensional ball for inequality (1.14), without writing its E h_A^β computation). Selected: drawn by its position in the list of cited rows, not because it looked wrong.

1 bound row(s) here have been checked against material for their cited source (1 drawn by position, 0 chosen because something already looked wrong). Across this whole ledger 70 row(s) have been drawn by position, against 691 rows that name a source, counted on 2026-09-27: 64 were checked against material for their cited sources and for 6 the cited source could not be read at all. 5 more were chosen for suspicion and checked; they are counted apart, because a set selected for suspicion carries no rate. A row not listed here has NOT been checked.