Digital shifts and the question of Dick, Goda and Suzuki

A digital shift adds a fixed digit to each digit of each coordinate, modulo q, and throws the carries away. In base 5, the coordinate 0.34 shifted by 0.21 becomes 0.00, not 0.10 as ordinary addition would give. Each coordinate gets its own shift digits, one per digit position.

Without carries, a shift sends every elementary box to another box of the same shape: the first d digits of a coordinate are only relabelled. So a net with one point in every box keeps that property, whatever the shift. A digital shift is the simplest of Owen's nested scrambles, which all preserve the net property (Owen, 1995). It is the digital counterpart of the random shift of Cranley and Patterson. Click a point to see its digits; click a shift digit to raise it by one.

The separation proofs never look at a single point. They look at two points in nearby cells and at the change of each digit between them. That change is an integer between −q and q, and its remainder modulo q is fixed by the cell step alone. Only two integers have that remainder: σ and σ − q, the second one when the digit wraps around. The proofs already take the worse of the two.

A shift adds the same constant to a digit at both points. The remainder of the change stays the same; the shift only decides at which cells the digit wraps, so which of the two values occurs. Since the bound covers both values, it holds for every shift. In the paper's terms, a shifted graph form has offsets Φn + γ with a constant γ, and every proof uses only differences, in which γ cancels.

Try it on the net 𝒫_2,q of page 1: shift its eighteen digits however you like, and the exact separation stays above the bound c(q) of the theorem. The table follows the closest pair of the shifted net digit by digit.

In the column net 𝒞_q of page 1 all three coordinates read the same second and third digit, so every point sits on the diagonal of its cell. A shift with equal digits in the three coordinates at these two positions keeps that structure and leaves the minimum distance exactly as it was. If only the second digits agree, the distance can change, but δN^1/3 stays above 1 − (7q − 3)/(√3 q²), close to the cube value 1.

If the second digits differ, the three coordinates no longer wrap at the same cells. The ring shows, for one step direction, the values of the shared digit at which each coordinate wraps: its wrap window. A shift turns each window by that coordinate's own shift digit. Where one coordinate wraps alone, the step along its axis shrinks, and the value falls to about 1/√3. A lower bound still holds for every shift, so the family stays well separated, only with a smaller constant.

The net 𝒲_p of page 3 beats the cube value 1 because its wraps are locked: z reads a digit ζ, and x and y both read 2ζ mod p, so z wraps only where x and y wrap too. A shift of the second digits by (t, t, t′) gives x and y the digit 2ζ + t − 2t′ after a translation of the cells. If t ≡ 2t′, this is the same lock. If t ≡ 2t′ + 1, the digit wraps one step earlier, which allows the same wrap patterns, so the proven bound still holds (for p ≥ 53).

Every other shift lets z wrap alone at some cells, and there the step along e_z shrinks to about (4ε, 4ε, 2ε). The map on the left tries every shift of the second digits for a small prime. That only these two lines keep the lock is a computed observation, not a theorem.

In bases 2 and 3 the nets of the paper come from a lift that adds three digits at a time, and their bounds come from finite certificates instead of the lattice picture. For these, the worst shift can be computed exactly. Take two points whose digits differ by d. A nonzero digit of weight b^k changes the coordinate by d_kb^k, or by (d_k − b)b^k when that digit wraps, and a shift can choose the wraps freely, digit by digit and coordinate by coordinate. The worst case μ(d) is the distance from d to the nearest sum of the numbers b^k+1, one for each wrapped digit.

So the smallest separation over all shifts is a minimum over the differences of points of Σ μ(Δ_c)², a minimum weight of a code. It drops to almost nothing when some difference reads 0…01…1 in every coordinate: a shift can put both points on a carry chain, such as 0.00111 and 0.01000 in base 2: their digits differ by 0.01111, their values by one unit in the last place. These carry chains are why the every-shift certificates of the paper vanish at some resolutions and appear at others.