Ang lists several different Hokkien equivalents for the Mandarin postposition 邊 (at the edge of, near), but I find his remarks somewhat confusing.
His text reads as follows:
source: Ang Ui-jin 洪惟仁: Taiwan lisu yudian 臺灣禮俗語典 [Dictionary of Taiwanese Etiquettes and Customs]. Taipei: Independence Evening 自立晚報 (1986), 260-262.台語【幵kiⁿ】、【墘kîⁿ】和【邊piⁿ】有程度之分。比如【鼎鉉tiáⁿ-kîⁿ】指鍋的邊緣，是鍋的一部分。【鼎邊tiáⁿ-piⁿ】指鍋的內壁，【鼎幵tiáⁿ-ki(sic: kiⁿ?)】則是灶上緊臨鼎的地方，在鼎之外。若用於其他地方，如溪、海，【墘kîⁿ】、【幵kiⁿ】、【邊piⁿ】也有遠近之分。【幵kiⁿ】和【邊piⁿ】家個「仔」，意思又有程度之差。比如【海幵hái-kiⁿ】指海灘、【海幵仔hái-kiⁿ-á】則稍為遠一點。【海邊hái-piⁿ】指海濱可以蓋房子的地方，【海邊仔hái-piⁿ-á】指離海較勁的地方。這樣看來，普通話一個「邊」自籠統包括臺語好幾個詞，由進而遠為【墘kîⁿ】、【幵kiⁿ】、【幵仔kiⁿ-á】、【邊仔piⁿ-á】、【邊piⁿ】。在【墘kîⁿ】和【邊piⁿ】中間如果距離很短，還可以細分為【幵幵也kîⁿ-kiⁿ--a】、【幵也kiⁿ--a】【邊也piⁿ--a】【邊piⁿ】。「也--a」讀隨前變調，和「仔á」意思不同，讀者把音標弄清楚了，細細體會，當可了解臺語的精密、巧妙。
[He then goes on talking about etymologies, why he chose the characters he did, that 【鉉kîⁿ】 and 【舷kîⁿ】 are actually one and the same word which is usually spelt 【墘kîⁿ】 and so on before returning to the semantics part:]茲依遠近關係舉例如下：
墘(kîⁿ) | 幵(kiⁿ) 幵仔(kiⁿ-á) 邊仔(piⁿ-á) 邊(piⁿ)
| 海幵 海幵仔 海邊仔 海邊
溪仔墘 | 溪仔幵 溪仔邊
港墘 | 港幵 港幵仔 港邊
(again I replaced his Romanization with POJ for typographical reasons)
To paraphrase, here's how I understood it: Hokkien knows three different basic words for "at the edge of/near" which are (from close to far): 墘(kîⁿ), 幵(kiⁿ) and 邊(piⁿ). 墘(kîⁿ) and 幵(kiⁿ) both mean the edge of something, but 墘(kîⁿ) tends to the inside and 幵(kiⁿ) to the outside (溪仔墘 would then mean "edge of the creek", as in the last few centimetres of water, while 溪仔幵 would be "edge of the creek" as in the first few centimetres of land). 邊(piⁿ) seems to correspond to English "close, near". Also, if the distance between 幵(kiⁿ) and 邊(piⁿ) is rather far, two more places between 幵(kiⁿ) and 邊(piⁿ) can be distinguished: 幵仔(kiⁿ-á) and 邊仔(piⁿ-á), which tend towards 幵(kiⁿ) and 邊(piⁿ) respectively.
However, I the only dictionary I can find which lists kiⁿ is the 台日大辭典 (it also lists kiⁿ-á and piⁿ-á btw). This leads me to think that kiⁿ is probably not used very often.
Also, Angs own cooking pot example doesn't seem to quite fit this pattern. He says 【鼎鉉tiáⁿ-kîⁿ】 means "the edge of the pot" (鍋的邊緣), 【鼎邊tiáⁿ-piⁿ】 its "inside wall" (鍋的內壁) and 【鼎鉉tiáⁿ-kiⁿ】 a place on the stove close to the pot on the stove (灶上緊臨鼎的地方). If I'm not mistaken, this would mean that 【鼎鉉tiáⁿ-kîⁿ】 and 【鼎邊tiáⁿ-piⁿ】 are pretty much the same, possibly with 【鼎鉉tiáⁿ-kîⁿ】 tending to the outside and 【鼎邊tiáⁿ-piⁿ】 to the inside of the pot, while 【鼎鉉tiáⁿ-kiⁿ】 is outside and merely near the pot. Or maybe 鉉(kîⁿ) is a different word than 幵(kiⁿ) after all?
Do you know more about the issue? Or maybe have a native speaker's feeling about it?